% % Testing file for REDUCE Special Functions Package % % Chris Cannam, ZIB Berlin % October 1992 -> Feb 1993 % (only some of the time, of course) % % Corrections and comments to neun@sc.zib-berlin.de % on savesfs; % just in case it's off for some reason off bfspace; % to provide more similarity between runs % with old & new bigfloats let {sinh (~x) => (exp(x) - exp (-x))/2 }; % this will improve some results % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 1. Bernoulli numbers % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= off rounded; procedure do!*one!*bern(x); write ("Bernoulli ", x, " is ", bernoulli x); do*one*bern do!*one!*bern(1); - 1 Bernoulli 1 is ------ 2 do!*one!*bern(2); 1 Bernoulli 2 is --- 6 do!*one!*bern(3); Bernoulli 3 is 0 do!*one!*bern(13); Bernoulli 13 is 0 do!*one!*bern(14); 7 Bernoulli 14 is --- 6 do!*one!*bern(300); Bernoulli 300 is ( - 186387899520485901199504534184815606618219184663590593751\ 8715320655775958174360523134990756922303410810482600528769\ 4796420210012184158790061643029553704608291464348079647177\ 3719535693514415158342483315425004774743357558499902912677\ 5186293388721514970183351129809976971603227633930434923843\ 9848295803115933725653985747628800282891676355700124156069\ 41367995702212211519561707046505473575241)/866054419230 do!*one!*bern(-2); Bernoulli -2 is bernoulli(-2) do!*one!*bern(0); Bernoulli 0 is 1 for n := 2 step 2 until 100 do do!*one!*bern n; 1 Bernoulli 2 is --- 6 - 1 Bernoulli 4 is ------ 30 1 Bernoulli 6 is ---- 42 - 1 Bernoulli 8 is ------ 30 5 Bernoulli 10 is ---- 66 - 691 Bernoulli 12 is -------- 2730 7 Bernoulli 14 is --- 6 - 3617 Bernoulli 16 is --------- 510 43867 Bernoulli 18 is ------- 798 - 174611 Bernoulli 20 is ----------- 330 854513 Bernoulli 22 is -------- 138 - 236364091 Bernoulli 24 is -------------- 2730 8553103 Bernoulli 26 is --------- 6 - 23749461029 Bernoulli 28 is ---------------- 870 8615841276005 Bernoulli 30 is --------------- 14322 - 7709321041217 Bernoulli 32 is ------------------ 510 2577687858367 Bernoulli 34 is --------------- 6 - 26315271553053477373 Bernoulli 36 is ------------------------- 1919190 2929993913841559 Bernoulli 38 is ------------------ 6 - 261082718496449122051 Bernoulli 40 is -------------------------- 13530 1520097643918070802691 Bernoulli 42 is ------------------------ 1806 - 27833269579301024235023 Bernoulli 44 is ---------------------------- 690 596451111593912163277961 Bernoulli 46 is -------------------------- 282 - 5609403368997817686249127547 Bernoulli 48 is --------------------------------- 46410 495057205241079648212477525 Bernoulli 50 is ----------------------------- 66 - 801165718135489957347924991853 Bernoulli 52 is ----------------------------------- 1590 29149963634884862421418123812691 Bernoulli 54 is ---------------------------------- 798 - 2479392929313226753685415739663229 Bernoulli 56 is --------------------------------------- 870 84483613348880041862046775994036021 Bernoulli 58 is ------------------------------------- 354 - 1215233140483755572040304994079820246041491 Bernoulli 60 is ------------------------------------------------ 56786730 12300585434086858541953039857403386151 Bernoulli 62 is ---------------------------------------- 6 - 106783830147866529886385444979142647942017 Bernoulli 64 is ----------------------------------------------- 510 1472600022126335654051619428551932342241899101 Bernoulli 66 is ------------------------------------------------ 64722 - 78773130858718728141909149208474606244347001 Bernoulli 68 is ------------------------------------------------- 30 1505381347333367003803076567377857208511438160235 Bernoulli 70 is --------------------------------------------------- 4686 - 5827954961669944110438277244641067365282488301844260429 Bernoulli 72 is ------------------------------------------------------------ 140100870 34152417289221168014330073731472635186688307783087 Bernoulli 74 is ---------------------------------------------------- 6 - 24655088825935372707687196040585199904365267828865801 Bernoulli 76 is ---------------------------------------------------------- 30 414846365575400828295179035549542073492199375372400483487 Bernoulli 78 is ----------------------------------------------------------- 3318 Bernoulli 80 is ( - 4603784299479457646935574969019046849794257872751288919656\ 867)/230010 1677014149185145836823154509786269900207736027570253414881613 Bernoulli 82 is --------------------------------------------------------------- 498 Bernoulli 84 is ( - 2024576195935290360231131160111731009989917391198090877281\ 083932477)/3404310 Bernoulli 86 is 660714619417678653573847847426261496277830686653388931761996983/ 6 Bernoulli 88 is ( - 1311426488674017507995511424019311843345750275572028644296\ 919890574047)/61410 Bernoulli 90 is 11790572790210827998841233512492150837752549496696471162315452\ 15727922535/272118 Bernoulli 92 is ( - 1295585948207537527989427828538576749659341483719435143023\ 316326829946247)/1410 Bernoulli 94 is 12208138065797444696073016794132012039585084152026966214362151\ 05284649447/6 Bernoulli 96 is ( - 2116004495972665130975977281098242336730439543890602341506\ 38733420050668349987259)/4501770 Bernoulli 98 is 67908260672905495624051117546403605607342195728504487509073961\ 249992947058239/6 Bernoulli 100 is ( - 945980378191221252952274330694937218727028415330669361333\ 85696204311395415197247711)/33330 on rounded; precision 100; 12 do!*one!*bern(1); Bernoulli 1 is - 0.5 do!*one!*bern(2); Bernoulli 2 is 0.1666666666666666666666666666666666666666666666666666666666666\ 666666666666666666666666666666666666667 do!*one!*bern(3); Bernoulli 3 is 0 do!*one!*bern(13); Bernoulli 13 is 0 do!*one!*bern(14); Bernoulli 14 is 1.166666666666666666666666666666666666666666666666666666666666\ 666666666666666666666666666666666666667 do!*one!*bern(300); Bernoulli 300 is - 2.15214997327998682971981737675608819857345657278380670184934\ 3925846796995824343474766557085976828531e+375 do!*one!*bern(-2); Bernoulli -2 is bernoulli(-2) do!*one!*bern(0); Bernoulli 0 is 1 do!*one!*bern(38); Bernoulli 38 is 4.883323189735931666666666666666666666666666666666666666666666\ 666666666666666666666666666666666666667e+14 do!*one!*bern(400); Bernoulli 400 is - 6.84694485580645336061625858231088359767823009718062574241478\ 1150311357197834589950328990573681886442e+549 % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 2. Gamma function % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= on rounded; off complex; precision 40; 100 algebraic procedure wg(x); write ("gamma (", x, ") ==> ", gamma x); wg algebraic procedure wp(x); write ("-- precision ", x, ", from ", precision(x)); wp wg (1/2); gamma (0.5) ==> 1.772453850905516027298167483341145182798 wg (3/2); gamma (1.5) ==> 0.8862269254527580136490837416705725913988 write ("sqrt(pi)/2 ==> ", sqrt(pi)/2); sqrt(pi)/2 ==> 0.8862269254527580136490837416705725913988 wp(10); -- precision 10, from 40 for x := 0 step 5 until 100 do << wg (1 + x/1000); wg (-1 - x/13); wp (8+floor(x/4)) >>; gamma (1) ==> 1 gamma (-1) ==> gamma(-1) -- precision 8, from 10 gamma (1.005) ==> 0.99713854 gamma ( - 1.3846154) ==> 2.7320314 -- precision 9, from 8 gamma (1.01) ==> 0.994325851 gamma ( - 1.76923077) ==> 2.89933597 -- precision 10, from 9 gamma (1.015) ==> 0.9915612888 gamma ( - 2.153846154) ==> - 2.919307224 -- precision 11, from 10 gamma (1.02) ==> 0.98884420326 gamma ( - 2.5384615385) ==> - 0.91247160689 -- precision 13, from 11 gamma (1.025) ==> 0.9861739631483 gamma ( - 2.923076923077) ==> - 2.407817725014 -- precision 14, from 13 gamma (1.03) ==> 0.98354995055382 gamma ( - 3.3076923076923) ==> 0.42665848359037 -- precision 15, from 14 gamma (1.035) ==> 0.980971560550586 gamma ( - 3.69230769230769) ==> 0.250121998146955 -- precision 16, from 15 gamma (1.04) ==> 0.9784382009142447 gamma ( - 4.076923076923077) ==> - 0.4868211588210416 -- precision 18, from 16 gamma (1.045) ==> 0.975949291822951489 gamma ( - 4.46153846153846154) ==> - 0.064315864992688343 -- precision 19, from 18 gamma (1.05) ==> 0.9735042655627756432 gamma ( - 4.846153846153846154) ==> - 0.07308480130893001268 -- precision 20, from 19 gamma (1.055) ==> 0.9711025662416699039 gamma ( - 5.2307692307692307692) ==> 0.026504298643014994546 -- precision 21, from 20 gamma (1.06) ==> 0.968743649511638364209 gamma ( - 5.61538461538461538462) ==> 0.00947958151841097406813 -- precision 23, from 21 gamma (1.065) ==> 0.96642698229883993296188 gamma (-6) ==> gamma(-6) -- precision 24, from 23 gamma (1.07) ==> 0.964152042541366448869499 gamma ( - 6.38461538461538461538462) ==> - 0.00224562428754660955672785 -- precision 25, from 24 gamma (1.075) ==> 0.9619183189344448686422338 gamma ( - 6.769230769230769230769231) ==> - 0.0014913552210721080976799 -- precision 26, from 25 gamma (1.08) ==> 0.95972531068282223532653464 gamma ( - 7.1538461538461538461538462) ==> 0.0009821350464726696000373901 -- precision 28, from 26 gamma (1.085) ==> 0.9575725272601010524841387298 gamma ( - 7.538461538461538461538461538) ==> 0.0002081367552774423831456972591 -- precision 29, from 28 gamma (1.09) ==> 0.95545948817480124076971838245 gamma ( - 7.9230769230769230769230769231) ==> 0.00038372470359721945415080451977 -- precision 30, from 29 gamma (1.095) ==> 0.95338572274293305256571929565 gamma ( - 8.30769230769230769230769230769) ==> - 0.0000487302698231202992921248405302 -- precision 31, from 30 gamma (1.1) ==> 0.9513507698668731836292487177265 gamma ( - 8.692307692307692307692307692308) ==> - 0.00002092711689267207910068115762837 -- precision 33, from 31 wg(1/1000000003); gamma (0.000000000999999997000000008999999973) ==> 1.00000000242278433608752313084681e+9 off rounded; gamma(17/2); 2027025*sqrt(pi) ------------------ 256 gamma(-17/2); - 512*pi ------------------- 34459425*sqrt(pi) gamma(4); 6 gamma(0); gamma(0) gamma(-4); gamma(-4) gamma(-17/3); 6*pi ------------------------ 17 17*sqrt(3)*gamma(----) 3 p := gamma(x**2) * gamma(x-y**gamma(y)) - (1/(gamma(4*(x-y)))); 2 gamma(y) gamma(x )*gamma( - y + x)*gamma(4*x - 4*y) - 1 p := -------------------------------------------------------- gamma(4*x - 4*y) y := 1/4; 1 y := --- 4 p; 2 1 gamma(1/4) gamma(x )*gamma( - (---) + x)*gamma(4*x - 1) - 1 4 ------------------------------------------------------------ gamma(4*x - 1) x := 3; x := 3 p; 1 gamma(1/4) 146313216000*gamma( - (---) + 3) - 1 4 ------------------------------------------------ 3628800 y := -3/8; - 3 y := ------ 8 p; (128*(2490343877896875*sqrt(pi) - 3 (8*pi)/(3*gamma(3/8)*sin((3*pi)/8)) 3*(------) - 1 8 *gamma(---------------------------------------------------) - 64))/( - 3 (8*pi)/(3*gamma(3/8)*sin((3*pi)/8)) (------) 8 7905853580625*sqrt(pi)) on rounded, complex; *** Domain mode rounded changed to complex-rounded precision 50; 33 p; - 0.00000000058461000084165968732153392127582134179078414159599 + 3.7721251013859508830301986850709684723938237902095e-60*i off rounded, complex; *** Domain mode complex-rounded changed to complex clear y; p; gamma(y) 40320*gamma( - y + 3)*gamma( - 4*y + 12) - 1 ------------------------------------------------------ gamma( - 4*y + 12) % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 3. Beta function. Not very interesting % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= algebraic procedure do!*one!*beta(x,y); write ("Beta (", x, ",", y, ") = ", beta(x,y)); do*one*beta do!*one!*beta(0,1); Beta (0,1) = beta(0,1) do!*one!*beta(2,-3); Beta (2,-3) = beta(2,-3) do!*one!*beta(3,2); 1 Beta (3,2) = ---- 12 do!*one!*beta(a+b,(c+d)**(b-a)); b (c + d) gamma(a + b)*gamma(----------) b a (c + d) (c + d) Beta (a + b,----------) = --------------------------------------------- a a a b (c + d) (c + d) *a + (c + d) *b + (c + d) gamma(------------------------------------) a (c + d) do!*one!*beta(-3,4); Beta (-3,4) = beta(-3,4) do!*one!*beta(-3,2); Beta (-3,2) = beta(-3,2) do!*one!*beta(-3,-7.5); - 15 - 15 Beta (-3,-------) = beta(-3,-------) 2 2 do!*one!*beta((pi * 10), exp(5)); 5 5 gamma(e )*gamma(10*pi) Beta (10*pi,e ) = ------------------------ 5 gamma(e + 10*pi) on rounded; precision 30; 50 do!*one!*beta(0,1); Beta (0,1) = beta(0,1) do!*one!*beta(2,-3); Beta (2,-3) = beta(2,-3) do!*one!*beta(3,2); Beta (3,2) = 0.0833333333333333333333333333333 do!*one!*beta(a+b,(c+d)**(b-a)); b (c + d) gamma(a + b)*gamma(----------) b a (c + d) (c + d) Beta (a + b,----------) = --------------------------------------------- a a a b (c + d) (c + d) *a + (c + d) *b + (c + d) gamma(------------------------------------) a (c + d) do!*one!*beta(-3,4); Beta (-3,4) = beta(-3,4) do!*one!*beta(-3,2); Beta (-3,2) = beta(-3,2) do!*one!*beta(-3,-7.5); Beta (-3, - 7.5) = beta(-3, - 7.5) do!*one!*beta((pi * 10), exp(5)); Beta (31.4159265358979323846264338328,148.413159102576603421115580041) = 3.26162024071771351768890966259e-37 % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 4. Pochhammer notation % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= off rounded; Pochhammer(4,5); 6720 Pochhammer(-4,5); 0 Pochhammer(4,-5); pochhammer(4,-5) Pochhammer(-4,-5); - 1 ------- 15120 Pochhammer(17/2,12); 157783444591397625 -------------------- 4096 Pochhammer(-17/2,12); - 516891375 -------------- 4096 Pochhammer(1/3,14)*Pochhammer(2/3,15); 148260333813553014031851192320000000 -------------------------------------- 68630377364883 q := Pochhammer(1/5,11)*Pochhammer(2/5,11)*Pochhammer(3/5,11)* Pochhammer(1-1/5,11)*Pochhammer(1,11)*Pochhammer(6/5,11)* Pochhammer(70/50,11)*Pochhammer(8/5,11)*Pochhammer(9/5,11); q := 3083594385972516709231715156294765932567301688416624945842705314534752626\ 0421323040989302952930905614937458388095527696384131072/12924697071141057\ 41986576081359316958696581423282623291015625 on complex; Pochhammer(a+b*i,c)*Pochhammer(a-b*i,c); pochhammer(a - i*b,c)*pochhammer(a + i*b,c) a := 2; a := 2 b := 3; b := 3 c := 5; c := 5 Pochhammer(a+b*i,c)*Pochhammer(a-b*i,c); 8950500 off complex; on rounded; Pochhammer(1/5,11)*Pochhammer(2/5,11)*Pochhammer(3/5,11)* Pochhammer(1-1/5,11)*Pochhammer(1,11)*Pochhammer(6/5,11)* Pochhammer(70/50,11)*Pochhammer(8/5,11)*Pochhammer(9/5,11); 2.38581559706937212593793381562e+67 q; 2.38581559706937212593793381562e+67 Pochhammer(pi,floor (pi**8)); 2.47253079057195612235973919163e+33625 Pochhammer(-pi,floor (pi**7)); 5.91750008101140123889628058136e+9185 Pochhammer(1.5,floor (pi**8)); 1.88808885937650373473836451368e+33619 % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 5. Digamma function % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= procedure do!*one!*psi(x); << precision (precision(0) + 4)$ write("Psi of ", x, " is ", psi(x) ) >> ; do*one*psi clear x, y; z := x * ((x+y)**2 + (x**y)); y 2 2 z := x*(x + x + 2*x*y + y ) off rounded; do!*one!*psi(3); - 2*euler_gamma + 3 Psi of 3 is ---------------------- 2 do!*one!*psi(pi); Psi of pi is psi(pi) do!*one!*psi(1.005); 201 1 Psi of ----- is psi(-----) + 200 200 200 do!*one!*psi(1.995); pi 1 199*cot(-----)*pi + 199*psi(-----) + 200 399 200 200 Psi of ----- is ------------------------------------------ 200 199 do!*one!*psi(74); Psi of 74 is ( - 136851726813476721146087646859200*euler_gamma + 667084944417653637854891458725877)/13685172681347672114608764\ 6859200 do!*one!*psi(-1/2); - 1 Psi of ------ is - 2*log(2) - euler_gamma + 2 2 do!*one!*psi(-3); ***** Psi undefined for nonpositive integer argument -3 do!*one!*psi(z); y 2 2 y 3 2 2 Psi of x*(x + x + 2*x*y + y ) is psi(x *x + x + 2*x *y + x*y ) on rounded; precision 100; 62 do!*one!*psi(3); Psi of 3 is 0.9227843350984671393934879099175975689578406640600764011942327651\ 1513227322233532906305293670825325048537 do!*one!*psi(pi); Psi of 3.141592653589793238462643383279502884197169399375105820974944592307816\ 40628620899862803482534211706798214809 is 0.9772133079420067332920694864061823436408346099943256380095232865318105924777\ 14131730207565436292873435576949 do!*one!*psi(1.005); Psi of 1.005 is - 0.569020911344382831688295690950028809267968870263469094699243\ 7771372094405771031990077045450115105840960011306543 do!*one!*psi(1.995); Psi of 1.995 is 0.419554603028108628651620272287243954692129976503405961275768\ 00237211077024989654159358528162438486780461247996589393 do!*one!*psi(74); Psi of 74 is 4.297293118804667639106350362560367137919643935000983615964648570\ 79142072638532815275329153012799334519059885201047316254 do!*one!*psi(-1/2); Psi of - 0.5 is 0.03648997397857652055902366700124443280684039533956589295287\ 2746128345029282945897851326282715415875401365590709051546051\ 66846 do!*one!*psi(-3); ***** Psi undefined for nonpositive integer argument -3 do!*one!*psi(z); y 2 2 y 3 2 2 Psi of x*(x + x + 2*x*y + y ) is psi(x *x + x + 2*x *y + x*y ) precision 15; 132 x := 8/3; x := 2.66666666666667 y := 7/1000; y := 0.007 do!*one!*psi(z); Psi of 21.74768766103287773 is 3.056340330052438423 on complex; *** Domain mode rounded changed to complex-rounded array psicomptest(10); psicomptest(1) := -13.0 + 2.0i; psicomptest(1) := - 13.0 + 2.0*i psicomptest(2) := -9.5; psicomptest(2) := - 9.5 psicomptest(3) := -3.0 + i; psicomptest(3) := - 3.0 + i psicomptest(4) := 8.0i; psicomptest(4) := 8.0*i psicomptest(5) := 3.0; psicomptest(5) := 3.0 psicomptest(6) := 4.0 + 2.0i; psicomptest(6) := 4.0 + 2.0*i psicomptest(7) := 7.0 + 4.0i; psicomptest(7) := 7.0 + 4.0*i psicomptest(8) := 9.0 + 16.0i; psicomptest(8) := 9.0 + 16.0*i psicomptest(9) := 10.0; psicomptest(9) := 10.0 psicomptest(10) := 15.0 + 5.0i; psicomptest(10) := 15.0 + 5.0*i for j:=1:10 do do!*one!*psi(psicomptest(j)); Psi of - 13.0 + 2.0*i is 2.6137588586149231277909 + 2.9946009556428566351564*i Psi of - 9.5 is 2.30300103429768637527259355 Psi of - 3.0 + i is 1.294650320622476977271878482722 + 2.87667404746858117413405079475*i Psi of 8.0*i is 2.080745674911800978012091156263302 + 1.6332963267948966192317863440681425*i Psi of 3.0 is 0.922784335098467139393487909917597568958 Psi of 4.0 + 2.0*i is 1.395360746143208295887639101232315589394657 + 0.516961128796076382232241496006733373069544*i Psi of 7.0 + 4.0*i is 2.0326956522302214192655943142503615623915246486 + 0.5510181566532114322789718358305813971403337745*i Psi of 9.0 + 16.0*i is 2.89681672499674156396524701196330813055662505786883 + 1.08235712929483567733811718225816618166171737283367*i Psi of 10.0 is 2.251752589066721107647456163885851537211808918028330369 Psi of 15.0 + 5.0*i is 2.7304638296862949658807696179355434065486393774434862562913 + 0.3319504266337825064013487331466704838924601236522898251895*i for j:=1:10 do do!*one!*psi(conj(psicomptest(j))); Psi of - 13.0 - 2.0*i is 2.61375885861492312779092681414576935243927815653319886872004667 - 2.99460095564285663515635556968177810390527354210169915597435706*i Psi of - 9.5 is 2.30300103429768637527259355084976605222629263212926398296056\ 0277036 Psi of - 3.0 - i is 1.2946503206224769772718784827219107224762629717635416232329897241189051 - 2.8766740474685811741340507947500004904456562664038166655750624843901542*i Psi of - 8.0*i is 2.080745674911800978012091156263301960447423157094657043233\ 56772730687510837 - 1.633296326794896619231786344068142496793658097596197866746\ 28197612585099252*i Psi of 3.0 is 0.92278433509846713939348790991759756895784066406007640119423276\ 51151322732223353 Psi of 4.0 - 2.0*i is 1.3953607461432082958876391012323155893946567716670922275697526053560369304559\ 129117 - 0.5169611287960763822322414960067333730695439777431470516513256070610078161450\ 6425866*i Psi of 7.0 - 4.0*i is 2.0326956522302214192655943142503615623915246486036687439734425046244698504734\ 1034449618 - 0.5510181566532114322789718358305813971403337745049891709724778015097808994965\ 80820485152*i Psi of 9.0 - 16.0*i is 2.8968167249967415639652470119633081305566250578688281426281739736662826184456\ 17485762055843 - 1.0823571292948356773381171822581661816617173728336735134819382305761766533865\ 22373036427982*i Psi of 10.0 is 2.2517525890667211076474561638858515372118089180283303694482010\ 190833862414763035830313069049622 Psi of 15.0 - 5.0*i is 2.7304638296862949658807696179355434065486393774434862562913209669772515134843\ 2633413224249878816897 - 0.3319504266337825064013487331466704838924601236522898251895017179559937376738\ 55349668081047744811948*i off complex, rounded; *** Domain mode complex-rounded changed to rounded clear x, y; df(psi(z), x); y 3 2 2 y y 2 2 polygamma(1,x *x + x + 2*x *y + x*y )*(x *y + x + 3*x + 4*x*y + y ) df(df(psi(z), y),x); 2*y y 3 2 2 x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x*y 2*y y 3 2 2 + x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x y y 3 2 2 3 + 3*x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x y y 3 2 2 2 + 4*x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x *y y y 3 2 2 2 + x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x*y y y 3 2 2 + x *log(x)*polygamma(1,x *x + x + 2*x *y + x*y )*y y y 3 2 2 + x *log(x)*polygamma(1,x *x + x + 2*x *y + x*y ) y y 3 2 2 2 + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x *y y y 3 2 2 2 + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x y y 3 2 2 2 + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x*y y y 3 2 2 + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x*y y y 3 2 2 + x *polygamma(1,x *x + x + 2*x *y + x*y ) y 3 2 2 4 + 6*polygamma(2,x *x + x + 2*x *y + x*y )*x y 3 2 2 3 + 14*polygamma(2,x *x + x + 2*x *y + x*y )*x *y y 3 2 2 2 2 + 10*polygamma(2,x *x + x + 2*x *y + x*y )*x *y y 3 2 2 3 + 2*polygamma(2,x *x + x + 2*x *y + x*y )*x*y y 3 2 2 + 4*polygamma(1,x *x + x + 2*x *y + x*y )*x y 3 2 2 + 2*polygamma(1,x *x + x + 2*x *y + x*y )*y int(psi(x), x); log(gamma(x)) on rounded; for k := 1 step 0.1 until 2 do do!*one!*psi(k); Psi of 1 is - 0.5772156649015328606065120900824024310421593359399235988057672348\ 848677267776646709369470632917467495146 Psi of 1.1 is - 0.42375494041107679516821621901002540429164256244418892032639208\ 410886791088152627023153983491219927980824243 Psi of 1.200000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000436 is - 0.2890398965921882955472079624499521048255882742066428101758586641916247540916\ 19652546577824319570362412407865259 Psi of 1.300000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000043598847 is - 0.1691908888667996556311611740386482662045749550711687623315864179308303857025\ 770257187015968591396298192251105235858 Psi of 1.400000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000435988468871 is - 0.0613845445851161457306754820475284558263610965108101572339536752126110429305\ 41383972160692353613817412775940509279641163 Psi of 1.500000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000004359884688711001 is 0.0364899739785765205590236670012444328068403953395658929528727461283450292829\ 4589785132628271541587540136563146534955733265 Psi of 1.600000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000043598846887110012947 is 0.1260474527734762519060027178604347251561324119770594583999814034054978254255\ 642005933241989468878465660903942111555360670073983 Psi of 1.700000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000435988468871100129469698 is 0.2085478748734939566799641840099706284504372410359405588951627233510439957197\ 1888196195835704807678107428159405980321416315399468575 Psi of 1.800000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000435988468871100129469697747 is 0.2849914332938615406087023668431645803658395103047771708901892057550387926143\ 15995693623340760077947867402943634838596190123409034342599 Psi of 1.900000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000043598846887110012946969774698648 is 0.3561841611640597192247270839286155915181509595109651018840831788470124953308\ 95055258791830992043449174689987285634836913342490624091145471 off rounded; % PSI_SIMP.TST F.J.Wright, 2 July 1993 on evallhseqp; factor psi; on rat, intstr, div; % for neater output % Do not try using "off mcd"! psi(x+m) - psi(x+m-1) = 1/(x+m-1); 1 1 -----------=----------- m + x - 1 m + x - 1 psi(x+2) - psi(x+1) + 2*psi(x) = 1/(x+1) + 2*psi(x); 1 1 2*psi(x) + -------=2*psi(x) + ------- x + 1 x + 1 psi(x+2) + 3*psi(x) = 4*psi(x) + 1/x + 1/(x+1); -1 -1 x + 2 x + 2 4*psi(x) + ---------=4*psi(x) + --------- x + 1 x + 1 psi(x + 1) = psi(x) + 1/x; -1 -1 psi(x) + x =psi(x) + x psi(x + 3/2) = psi(x + 1/2) + 1/(x + 1/2); 1 2 1 2 psi(x + ---) + ---------=psi(x + ---) + --------- 2 2*x + 1 2 2*x + 1 psi(x - 1/2) = psi(x + 1/2) - 1/(x - 1/2); 1 2 1 2 psi(x + ---) - ---------=psi(x + ---) - --------- 2 2*x - 1 2 2*x - 1 psi((x + 3a)/a); -1 1 2*(3*x + 8*x + 12) psi(---*x) + ---------------------- 2 2 x + 6*x + 8 psi(x/y + 3); -1 2 -1 y*(3*x + 2*x *y + 6*y) psi(x*y ) + -------------------------- 2 2 x + 3*x*y + 2*y off rat, intstr, div; on rational; psi(x+m) - psi(x+m-1) = 1/(x+m-1); 1 1 -----------=----------- m + x - 1 m + x - 1 psi(x+2) - psi(x+1) + 2*psi(x) = 1/(x+1) + 2*psi(x); 2*psi(x)*x + 2*psi(x) + 1 2*psi(x)*x + 2*psi(x) + 1 ---------------------------=--------------------------- x + 1 x + 1 psi(x+2) + 3*psi(x) = 4*psi(x) + 1/x + 1/(x+1); 2 2 4*psi(x)*x + 4*psi(x)*x + 2*x + 1 4*psi(x)*x + 4*psi(x)*x + 2*x + 1 ------------------------------------=------------------------------------ 2 2 x + x x + x psi(x + 1) = psi(x) + 1/x; psi(x)*x + 1 psi(x)*x + 1 --------------=-------------- x x psi(x + 3/2) = psi(x + 1/2) + 1/(x + 1/2); 1 1 1 1 1 1 psi(x + ---)*x + ---*psi(x + ---) + 1 psi(x + ---)*x + ---*psi(x + ---) + 1 2 2 2 2 2 2 ---------------------------------------=--------------------------------------- 1 1 x + --- x + --- 2 2 psi(x - 1/2) = psi(x + 1/2) - 1/(x - 1/2); 1 1 1 1 1 1 psi(x + ---)*x - ---*psi(x + ---) - 1 psi(x + ---)*x - ---*psi(x + ---) - 1 2 2 2 2 2 2 ---------------------------------------=--------------------------------------- 1 1 x - --- x - --- 2 2 psi((x + 3a)/a); 1 2 2 8 psi(---*x)*x*(x + 6*x + 8) + 6*(x + 4*x + ---) 2 3 -------------------------------------------------- 2 x*(x + 6*x + 8) psi(x/y + 3); x 2 2 2 2 2 psi(---)*x*(x + 3*x*y + 2*y ) + 3*y*(x + 2*x*y + ---*y ) y 3 ------------------------------------------------------------ 2 2 x*(x + 3*x*y + 2*y ) off rational; % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 6. Polygamma functions % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= procedure do!*one!*pg(n,x); write ("Polygamma (", n, ") of ", x, " is ", polygamma(n,x)); do*one*pg off rounded; do!*one!*pg(1,1/2); 1 1 Polygamma (1) of --- is polygamma(1,---) 2 2 do!*one!*pg(1,1); 2 pi Polygamma (1) of 1 is ----- 6 do!*one!*pg(1,3/2); 2 3 pi - 8 Polygamma (1) of --- is --------- 2 2 do!*one!*pg(1,1.005); 201 201 Polygamma (1) of ----- is polygamma(1,-----) 200 200 do!*one!*pg(1,1.995); 399 399 Polygamma (1) of ----- is polygamma(1,-----) 200 200 do!*one!*pg(1,1e-10); 1 1 Polygamma (1) of ------------- is polygamma(1,-------------) 10000000000 10000000000 do!*one!*pg(2,1.45); 29 29 Polygamma (2) of ---- is polygamma(2,----) 20 20 do!*one!*pg(3,1.99); 199 199 Polygamma (3) of ----- is polygamma(3,-----) 100 100 do!*one!*pg(4,-8.2); - 41 - 41 Polygamma (4) of ------- is polygamma(4,-------) 5 5 do!*one!*pg(5,0); ***** Polygamma undefined for nonpositive integer argument 0 do!*one!*pg(6,-5); ***** Polygamma undefined for nonpositive integer argument -5 do!*one!*pg(7,200); Polygamma (7) of 200 is (17726903568079021516229662669344581999513453462651624\ 712682433450930396569413638060869523209196416456149981841411350319061723276\ 968944831274405326841106087088051862580482604049179034792880997036306450715\ 491134920388121743532178874504889199026667348488846246287054278592284199023\ 548572837863642592371308428279823642858774260678182876022899303888558378920\ 013392172441409952343098219076353830605697314645020537255667718925997023946\ 826863459069360054494011515187931095594830347819766125471751832158677864778\ 297413490571816668099535649328289203181141151677544880676008798702466775519\ 536091686976761627732998715124407667063278175501392084867643186474774933465\ 8 396962161891279302009565617687762042880000000000000000000000*pi - 16820227\ 432460991991188660317955383248583163197077694391321472430675801147397894113\ 241309474829652161707416711766105349898340694530697667469860241947942732920\ 236500114448913470876512035623237448423938602084906142995199152247445505854\ 809437866123469569241946248640222536901257402027830808909074139755120553020\ 556624279638002199727273508802505439369471450464398635245844612858514905584\ 191236791869017030402868269742260341406479951357060228410068248515181595142\ 787674466216948716753957289698597496283251496652438121703517099395342367021\ 250291313681953557646793694608947926057983579614775135882107369446401248554\ 435651401852361884349691269283596626501141334161214962487130496595028983095\ 5139972163546512706990894178649281)/332379441901481653429306175050210912490\ 877252424717963362795627204944935676505713641303560172432808552812159526462818\ 482407311443167715586395099878270739132900972423384048825922106902366518694430\ 745950915458779757277282691228353896966672481750012784165867117882267723605328\ 731691535740709943298606962033030246693303602017387715928925429361947910469604\ 750251103233276436606433091607681634323856824649594135073543769729862444199003\ 003689857550501021762715909773708042403069021620614852595346852975209964593076\ 502948221562526866293424905422559646396593953966512675164975671252040991301719\ 130814280519993725908582643757436465790651101591268309746402030002476193040535\ 46148691267935533164553830400000000000000000000000 on rounded; precision 100; 139 do!*one!*pg(1,1/2); Polygamma (1) of 0.5 is 4.9348022005446793094172454999380755676568497036203953132066746881100224112096\ 02621500886701859276116 do!*one!*pg(1,1); Polygamma (1) of 1 is 1.6449340668482264364724151666460251892189499012067984377355582293700074704032\ 00873833628900619758705 do!*one!*pg(1,3/2); Polygamma (1) of 1.5 is 0.9348022005446793094172454999380755676568497036203953132066746881100224112096\ 026215008867018592761159 do!*one!*pg(1,1.005); Polygamma (1) of 1.005 is 1.6329941567556809752535869786110269899606654266292810174237683064726679207401\ 36690588658095268582677 do!*one!*pg(1,1.995); Polygamma (1) of 1.995 is 0.6469608286405823512476399271078276952436864784225077824256922477813603546774\ 633500406650324443323374 do!*one!*pg(1,1e-10); Polygamma (1) of 0.0000000001 is 1.0000000000000000000164493406660781505587296600657326399164207520577564247801\ 80942170336642993019363e+20 do!*one!*pg(2,1.45); Polygamma (2) of 1.45 is - 0.9038374030762576882323095385298671928600903081114466582453675225591714826271\ 66928803699348863802546 do!*one!*pg(3,1.99); Polygamma (3) of 1.99 is 0.5029071324168653201805109908625701971861516895613080444480934082942919777908\ 933506993776280887706183 do!*one!*pg(4,-8.2); Polygamma (4) of - 8.2 is 74935.512595774270527120292307813135522959054934662779176176297186107113915939\ 37121099755096024434196 do!*one!*pg(5,0); ***** Polygamma undefined for nonpositive integer argument 0 do!*one!*pg(6,-5); ***** Polygamma undefined for nonpositive integer argument -5 do!*one!*pg(7,200); Polygamma (7) of 200 is 5.7240937253925583738370296815447272453468073681959561547435678473859266441297\ 53317064645661270233713e-14 off rounded; clear x; % Polygamma differentiation has already % been tried a bit in the psi section df(int(int(int(polygamma(3,x),x),x),x),x); polygamma(1,x) clear w, y, z; % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 7. Zeta function % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= procedure do!*one!*zeta(n); write ("Zeta of ", n, " is ", zeta n); do*one*zeta off rounded; clear x, y, z; z := x * ((x+y)**5 + (x**y)); y 5 4 3 2 2 3 4 5 z := x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x*y + y ) do!*one!*zeta(0); - 1 Zeta of 0 is ------ 2 for k := 4 step 2 until 35 do do!*one!*zeta(k); 4 pi Zeta of 4 is ----- 90 6 pi Zeta of 6 is ----- 945 8 pi Zeta of 8 is ------ 9450 10 pi Zeta of 10 is ------- 93555 12 691*pi Zeta of 12 is ----------- 638512875 14 2*pi Zeta of 14 is ---------- 18243225 16 3617*pi Zeta of 16 is -------------- 325641566250 18 43867*pi Zeta of 18 is ---------------- 38979295480125 20 174611*pi Zeta of 20 is ------------------ 1531329465290625 22 155366*pi Zeta of 22 is ------------------- 13447856940643125 24 236364091*pi Zeta of 24 is ----------------------- 201919571963756521875 26 1315862*pi Zeta of 26 is ---------------------- 11094481976030578125 28 6785560294*pi Zeta of 28 is -------------------------- 564653660170076273671875 30 6892673020804*pi Zeta of 30 is ------------------------------ 5660878804669082674070015625 Zeta of 32 is zeta(32) Zeta of 34 is zeta(34) for k := 1 step 2 until 31 do do!*one!*zeta(-k); - 1 Zeta of -1 is ------ 12 1 Zeta of -3 is ----- 120 - 1 Zeta of -5 is ------ 252 1 Zeta of -7 is ----- 240 - 1 Zeta of -9 is ------ 132 691 Zeta of -11 is ------- 32760 - 1 Zeta of -13 is ------ 12 3617 Zeta of -15 is ------ 8160 - 43867 Zeta of -17 is ---------- 14364 174611 Zeta of -19 is -------- 6600 - 77683 Zeta of -21 is ---------- 276 236364091 Zeta of -23 is ----------- 65520 - 657931 Zeta of -25 is ----------- 12 3392780147 Zeta of -27 is ------------ 3480 - 1723168255201 Zeta of -29 is ------------------ 85932 Zeta of -31 is zeta(-31) do!*one!*zeta(-17/3); - 17 - 17 Zeta of ------- is zeta(-------) 3 3 do!*one!*zeta(190); Zeta of 190 is zeta(190) do!*one!*zeta(300); Zeta of 300 is zeta(300) do!*one!*zeta(0); - 1 Zeta of 0 is ------ 2 do!*one!*zeta(-44); Zeta of -44 is 0 on rounded; clear x, y; for k := 3 step 3 until 36 do << precision (31+k*3); do!*one!*zeta(k) >>; Zeta of 3 is 1.202056903159594285399738161511449990765 Zeta of 6 is 1.017343061984449139714517929790920527901817490033 Zeta of 9 is 1.002008392826082214417852769232412060485605851394888756549 Zeta of 12 is 1.00024608655330804829863799804773967096041608845800340453304095\ 2133 Zeta of 15 is 1.00003058823630702049355172851064506258762794870685817750656993\ 2893332267156 Zeta of 18 is 1.00000381729326499983985646164462193973045469721895333114317442\ 9987630039542650045638 Zeta of 21 is 1.00000047693298678780646311671960437304596644669478493760020748\ 7376596839087898159833876638564 Zeta of 24 is 1.00000005960818905125947961244020793580122750391883730279586424\ 6972321724495355468544848206832825003614 Zeta of 27 is 1.00000000745071178983542949198100417060411945471903188256582999\ 3239578352147606271570867900837100313523764933952 Zeta of 30 is 1.00000000093132743241966818287176473502121981356795513681618500\ 8613360441960672940496363503624604027929086312123388047291 Zeta of 33 is 1.00000000011641550172700519775929738354563095165224717276359325\ 6517739947029124624567548673934974376008810870912845774213829513\ 369 Zeta of 36 is 1.00000000001455192189104198423592963224531842098380889412403806\ 9139542218571745865030220152998942329578185363084791339999779092\ 891491916899 precision 20; 139 do!*one!*zeta(-17/3); Zeta of - 5.6666666666666666667 is - 0.0018766468228592287697 do!*one!*zeta(z); y 5 4 3 2 2 3 4 5 Zeta of x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x*y + y ) is zeta( y 6 5 4 2 3 3 2 4 5 x *x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + x*y ) y := 3; y := 3 x := pi; x := 3.1415926535897932385 do!*one!*zeta(z); Zeta of 27548.203250393209469 is 1 do!*one!*zeta(190); Zeta of 190 is 1.0 do!*one!*zeta(300); Zeta of 300 is 1 do!*one!*zeta(0); Zeta of 0 is - 0.5 do!*one!*zeta(-44); Zeta of -44 is 0 off rounded; % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 8. Kummer functions % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= off rounded; t!*kummer!*a := { 1, 2.4, -1397/10 }$ t!*kummer!*b := { 0, 1, pi, -pi, 26 }$ for each a in t!*kummer!*a do for each b in t!*kummer!*a do for each z in t!*kummer!*a do << write "KummerM(", a, ",", b, ",", z, ") = ", KummerM(a,b,z); write "KummerU(", a, ",", b, ",", z, ") = ", KummerU(a,b,z) >>; KummerM(1,1,1) = e KummerU(1,1,1) = kummeru(1,1,1) 12 2/5 2 KummerM(1,1,----) = e *e 5 12 12 KummerU(1,1,----) = kummeru(1,1,----) 5 5 - 1397 1 KummerM(1,1,---------) = ------------ 10 7/10 139 e *e - 1397 - 1397 KummerU(1,1,---------) = kummeru(1,1,---------) 10 10 12 12 KummerM(1,----,1) = kummerm(1,----,1) 5 5 2 12 12 2*gamma(---)*(gamma(----)*e - kummerm(1,----,1)) 12 5 5 5 KummerU(1,----,1) = -------------------------------------------------- 5 12 5*gamma(----) 5 12 12 12 12 KummerM(1,----,----) = kummerm(1,----,----) 5 5 5 5 12 12 KummerU(1,----,----) = 5 5 2 2/5 2/5 12 2 4/5 2/5 12 12 gamma(---)*(5*e *5 *gamma(----)*e - 12*2 *3 *kummerm(1,----,----)) 5 5 5 5 ----------------------------------------------------------------------------- 4/5 2/5 12 30*2 *3 *gamma(----) 5 12 - 1397 12 - 1397 KummerM(1,----,---------) = kummerm(1,----,---------) 5 10 5 10 12 - 1397 2 KummerU(1,----,---------) = (2*gamma(---)*( 5 10 5 7/10 2/5 12 - 1397 139 - 1397*e *1397 *kummerm(1,----,---------)*e 5 10 2/5 12 7/10 2/5 12 139 - 10*10 *gamma(----)))/(6985*e *1397 *gamma(----)*e ) 5 5 - 1397 - 1397 KummerM(1,---------,1) = kummerm(1,---------,1) 10 10 - 1397 KummerU(1,---------,1) = 10 1397 - 1397 3*pi 1397*gamma(------)*kummerm(1,---------,1)*sin(------) - 10*e*pi 10 10 10 ----------------------------------------------------------------- 1417 3*pi 10*gamma(------)*sin(------) 10 10 - 1397 12 - 1397 12 KummerM(1,---------,----) = kummerm(1,---------,----) 10 5 10 5 - 1397 12 KummerU(1,---------,----) = ( - 4868940170769807140163734629614870951763530546\ 10 5 631554678436906268603895538747067202821442117249242625797089485978560806001\ 2/5 2/5 7/10 2 0798787187515790195560655355904*e *2 *3 *e *pi + 2004596689573651914\ 834381033324557143448090555188728353940095378016295057932438794523477554321\ 7/10 1397 - 1397 12 3*pi 2890625*5 *gamma(------)*kummerm(1,---------,----)*sin(------))/(1434929\ 10 10 5 10 627468612680625899093288874118430988228481552150279237922703146068670321255\ 7/10 1417 3*pi 92231750488281250*5 *gamma(------)*sin(------)) 10 10 - 1397 - 1397 - 1397 - 1397 KummerM(1,---------,---------) = kummerm(1,---------,---------) 10 10 10 10 - 1397 - 1397 KummerU(1,---------,---------) = (1397*( - 15215998257931517360033101903862815\ 10 10 223888543388957222106115555299202666279048867916168002482075600873075777\ 814303704269392393284644411305647344643215427879407971076364582034085385\ 160268928702508481521778353248095651747223572373118125240197785695254459\ 776199438476999663624567921028135476855359579600382069871166959862260275\ 271370315144729355759279591209107609434598035344309681771044690476129730\ 7/10 6235868715700448137787799653833690839289133*( - 1397) *pi + 100000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 7/10 7/10 00000000000000000000000000000000000000000000000000000000000*e *10 1397 - 1397 - 1397 3*pi 139 *gamma(------)*kummerm(1,---------,---------)*sin(------)*e ))/(100000\ 10 10 10 10 000000000000000000000000000000000000000000000000000000000000000000000000000\ 7/10 7/10 000000000000000000000000000000000000000000000000000000000000*e *10 1417 3*pi 139 *gamma(------)*sin(------)*e ) 10 10 12 12 KummerM(----,1,1) = kummerm(----,1,1) 5 5 12 12 KummerU(----,1,1) = kummeru(----,1,1) 5 5 12 12 12 12 KummerM(----,1,----) = kummerm(----,1,----) 5 5 5 5 12 12 12 12 KummerU(----,1,----) = kummeru(----,1,----) 5 5 5 5 12 - 1397 12 - 1397 KummerM(----,1,---------) = kummerm(----,1,---------) 5 10 5 10 12 - 1397 12 - 1397 KummerU(----,1,---------) = kummeru(----,1,---------) 5 10 5 10 12 12 KummerM(----,----,1) = e 5 5 2 - 2 2*pi 2*gamma(---)*kummerm(1,------,1)*sin(------) + 5*e*pi 12 12 5 5 5 KummerU(----,----,1) = ------------------------------------------------------- 5 5 12 2*pi 5*gamma(----)*sin(------) 5 5 12 12 12 2/5 2 KummerM(----,----,----) = e *e 5 5 5 12 12 12 KummerU(----,----,----) = 5 5 5 2/5 4/5 2/5 2 2/5 2 - 2 12 2*pi 6*e *2 *3 *e *pi + 5 *gamma(---)*kummerm(1,------,----)*sin(------) 5 5 5 5 ----------------------------------------------------------------------------- 4/5 2/5 12 2*pi 6*2 *3 *gamma(----)*sin(------) 5 5 12 12 - 1397 1 KummerM(----,----,---------) = ------------ 5 5 10 7/10 139 e *e 12 12 - 1397 KummerU(----,----,---------) = ( 5 5 10 7/10 2/5 2 - 2 - 1397 2*pi 139 - 4*e *10 *gamma(---)*kummerm(1,------,---------)*sin(------)*e 5 5 10 5 2/5 7/10 2/5 12 2*pi 139 + 1397*1397 *pi)/(1397*e *1397 *gamma(----)*sin(------)*e ) 5 5 12 - 1397 12 - 1397 KummerM(----,---------,1) = kummerm(----,---------,1) 5 10 5 10 12 - 1397 1431 1431 1417 KummerU(----,---------,1) = ( - 10*gamma(------)*kummerm(------,------,1)*pi + 5 10 10 10 10 1417 1397 12 12 - 1397 1397*gamma(------)*gamma(------)*gamma(----)*kummerm(----,---------,1) 10 10 5 5 10 3*pi 1431 1417 12 3*pi *sin(------))/(10*gamma(------)*gamma(------)*gamma(----)*sin(------)) 10 10 10 5 10 12 - 1397 12 12 - 1397 12 KummerM(----,---------,----) = kummerm(----,---------,----) 5 10 5 5 10 5 12 - 1397 12 KummerU(----,---------,----) = (2004596689573651914834381033324557143448090555\ 5 10 5 7/10 1417 1887283539400953780162950579324387945234775543212890625*5 *gamma(------) 10 1397 12 12 - 1397 12 3*pi *gamma(------)*gamma(----)*kummerm(----,---------,----)*sin(------) - 48689\ 10 5 5 10 5 10 401707698071401637346296148709517635305466315546784369062686038955387470672\ 2/5 028214421172492426257970894859785608060010798787187515790195560655355904*2 7/10 1431 1431 1417 12 *3 *gamma(------)*kummerm(------,------,----)*pi)/(143492962746861268062\ 10 10 10 5 589909328887411843098822848155215027923792270314606867032125592231750488281\ 7/10 1431 1417 12 3*pi 250*5 *gamma(------)*gamma(------)*gamma(----)*sin(------)) 10 10 5 10 12 - 1397 - 1397 12 - 1397 - 1397 KummerM(----,---------,---------) = kummerm(----,---------,---------) 5 10 10 5 10 10 12 - 1397 - 1397 KummerU(----,---------,---------) = (1397*( - 15215998257931517360033101903862\ 5 10 10 815223888543388957222106115555299202666279048867916168002482075600873075\ 777814303704269392393284644411305647344643215427879407971076364582034085\ 385160268928702508481521778353248095651747223572373118125240197785695254\ 459776199438476999663624567921028135476855359579600382069871166959862260\ 275271370315144729355759279591209107609434598035344309681771044690476129\ 7/10 1431 7306235868715700448137787799653833690839289133*( - 1397) *gamma(------) 10 1431 1417 - 1397 *kummerm(------,------,---------)*pi + 100000000000000000000000000000000\ 10 10 10 000000000000000000000000000000000000000000000000000000000000000000000000\ 7/10 1417 1397 00000000000000000000000000000000000*10 *gamma(------)*gamma(------) 10 10 12 12 - 1397 - 1397 3*pi *gamma(----)*kummerm(----,---------,---------)*sin(------)))/(1000000000\ 5 5 10 10 10 000000000000000000000000000000000000000000000000000000000000000000000000000\ 7/10 1431 00000000000000000000000000000000000000000000000000000000*10 *gamma(------) 10 1417 12 3*pi *gamma(------)*gamma(----)*sin(------)) 10 5 10 - 1397 - 1397 KummerM(---------,1,1) = kummerm(---------,1,1) 10 10 - 1397 - 1397 KummerU(---------,1,1) = kummeru(---------,1,1) 10 10 - 1397 12 - 1397 12 KummerM(---------,1,----) = kummerm(---------,1,----) 10 5 10 5 - 1397 12 - 1397 12 KummerU(---------,1,----) = kummeru(---------,1,----) 10 5 10 5 - 1397 - 1397 - 1397 - 1397 KummerM(---------,1,---------) = kummerm(---------,1,---------) 10 10 10 10 - 1397 - 1397 - 1397 - 1397 KummerU(---------,1,---------) = kummeru(---------,1,---------) 10 10 10 10 - 1397 12 - 1397 12 KummerM(---------,----,1) = kummerm(---------,----,1) 10 5 10 5 - 1397 12 KummerU(---------,----,1) = ( 10 5 1411 - 1397 12 pi 7055*gamma(------)*kummerm(---------,----,1)*sin(----)*pi + 2794 10 10 5 10 1397 12 2 - 1411 - 2 3*pi *gamma(------)*gamma(----)*gamma(---)*kummerm(---------,------,1)*sin(------) 10 5 5 10 5 10 2*pi 12 2*pi *sin(------))/(50*gamma(----)*sin(------)*pi) 5 5 5 - 1397 12 12 - 1397 12 12 KummerM(---------,----,----) = kummerm(---------,----,----) 10 5 5 10 5 5 - 1397 12 12 2/5 1397 12 2 KummerU(---------,----,----) = (1397*5 *gamma(------)*gamma(----)*gamma(---) 10 5 5 10 5 5 - 1411 - 2 12 3*pi 2*pi *kummerm(---------,------,----)*sin(------)*sin(------) 10 5 5 10 5 4/5 2/5 1411 - 1397 12 12 pi + 8466*2 *3 *gamma(------)*kummerm(---------,----,----)*sin(----)*pi)/( 10 10 5 5 10 4/5 2/5 12 2*pi 60*2 *3 *gamma(----)*sin(------)*pi) 5 5 - 1397 12 - 1397 - 1397 12 - 1397 KummerM(---------,----,---------) = kummerm(---------,----,---------) 10 5 10 10 5 10 - 1397 12 - 1397 KummerU(---------,----,---------) = ( 10 5 10 2/5 1411 - 1397 12 - 1397 pi 1411*1397 *gamma(------)*kummerm(---------,----,---------)*sin(----)*pi - 4 10 10 5 10 10 2/5 1397 12 2 *10 *gamma(------)*gamma(----)*gamma(---) 10 5 5 - 1411 - 2 - 1397 3*pi 2*pi 2/5 *kummerm(---------,------,---------)*sin(------)*sin(------))/(10*1397 10 5 10 10 5 12 2*pi *gamma(----)*sin(------)*pi) 5 5 - 1397 - 1397 KummerM(---------,---------,1) = e 10 10 - 1397 - 1397 KummerU(---------,---------,1) = 10 10 1397 1417 1417 1397*gamma(------)*(gamma(------)*e - kummerm(1,------,1)) 10 10 10 ------------------------------------------------------------ 1417 10*gamma(------) 10 - 1397 - 1397 12 2/5 2 KummerM(---------,---------,----) = e *e 10 10 5 - 1397 - 1397 12 1397 KummerU(---------,---------,----) = (1397*gamma(------)*(717464813734306340312\ 10 10 5 10 949546644437059215494114240776075139618961351573034335160627961158752441\ 2/5 7/10 1417 2 40625*e *5 *gamma(------)*e - 243447008538490357008186731480743547\ 10 588176527331577733921845313430194776937353360141072105862462131289854474\ 2/5 7/10 29892804030005399393593757895097780327677952*2 *3 1417 12 *kummerm(1,------,----)))/(717464813734306340312949546644437059215494114\ 10 5 7/10 1417 240776075139618961351573034335160627961158752441406250*5 *gamma(------)) 10 - 1397 - 1397 - 1397 1 KummerM(---------,---------,---------) = ------------ 10 10 10 7/10 139 e *e - 1397 - 1397 - 1397 1397 KummerU(---------,---------,---------) = (1397*gamma(------)*( - 2125674956633\ 10 10 10 10 032975196624335969635286777229511437323928224343075298612479183126847888\ 669946745961441968686160658227486434117341864824259398934046657195274753\ 293559368132110161728306889569339740434868592435948758962549087133060524\ 602096055630661627048030735061555236853008352138567630526116693733270173\ 375161002024292757760455410433025718690999571358891912333038013345537600\ 7/10 06254341494325951532336811508595833526048489556116405666102486918801*e 7/10 1417 - 1397 139 *( - 1397) *kummerm(1,------,---------)*e + 1000000000000000000000\ 10 10 000000000000000000000000000000000000000000000000000000000000000000000000\ 7/10 1417 00000000000000000000000000000000000000000000000*10 *gamma(------)))/( 10 100000000000000000000000000000000000000000000000000000000000000000000000000\ 7/10 0000000000000000000000000000000000000000000000000000000000000000000*e 7/10 1417 139 *10 *gamma(------)*e ) 10 on rounded; precision 30; 20 t!*k!*c := 7; t*k*c := 7 % To test each and every possible combination of % three arguments from t!*kummer!*b would take too % long, but we want the possibility of trying most % special cases. Compromise: test every seventh % possibility. for each a in t!*kummer!*b do for each b in t!*kummer!*b do for each z in t!*kummer!*b do << if t!*k!*c = 7 then << write "KummerM(", a, ",", b, ",", z, ") = ", KummerM(a,b,z); write "KummerU(", a, ",", b, ",", z, ") = ", KummerU(a,b,z); t!*k!*c := 0 >>; t!*k!*c := t!*k!*c + 1 >>; KummerM(0,0,0) = kummerm(0,0,0) KummerU(0,0,0) = kummeru(0,0,0) KummerM(0,1,3.14159265358979323846264338328) = 1 KummerU(0,1,3.14159265358979323846264338328) = kummeru(0,1, 3.14159265358979323846264338328) KummerM(0,3.14159265358979323846264338328,26) = 1 KummerU(0,3.14159265358979323846264338328,26) = kummeru(0, 3.14159265358979323846264338328,26) KummerM(0,26,1) = 1 KummerU(0,26,1) = kummeru(0,26,1) KummerM(1,0, - 3.14159265358979323846264338328) = kummerm(1,0, - 3.14159265358979323846264338328) KummerU(1,0, - 3.14159265358979323846264338328) = kummeru(1,0, - 3.14159265358979323846264338328) KummerM(1,3.14159265358979323846264338328,0) = 1 KummerU(1,3.14159265358979323846264338328,0) = kummeru(1, 3.14159265358979323846264338328,0) KummerM(1, - 3.14159265358979323846264338328,3.14159265358979323846264338328) = 2692.89480079631357528203659153 KummerU(1, - 3.14159265358979323846264338328,3.14159265358979323846264338328) = 0.129554194296952806409644905659 KummerM(1,26,26) = 7.74565667206271943920803547594 KummerU(1,26,26) = kummeru(1,26,26) KummerM(3.14159265358979323846264338328,1,1) = 10.2259987795328570162092950355 KummerU(3.14159265358979323846264338328,1,1) = kummeru( 3.14159265358979323846264338328,1,1) KummerM(3.14159265358979323846264338328,3.14159265358979323846264338328, - 3.14159265358979323846264338328) = 0.0432139182637722497744177371717 KummerU(3.14159265358979323846264338328,3.14159265358979323846264338328, - 3.14159265358979323846264338328) = ( - 0.137891580772667438638357954721 2.14159265358979323846264338328 *( - 3.14159265358979323846264338328) + 1.22938452238615186004844111454)/ 2.14159265358979323846264338328 ( - 3.14159265358979323846264338328) KummerM(3.14159265358979323846264338328,26,0) = 1 KummerU(3.14159265358979323846264338328,26,0) = kummeru( 3.14159265358979323846264338328,26,0) KummerM( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) = kummerm( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) KummerU( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) = kummeru( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) KummerM( - 3.14159265358979323846264338328,1,26) = 6.18522226564722800173513559462e+5 KummerU( - 3.14159265358979323846264338328,1,26) = kummeru( - 3.14159265358979323846264338328,1,26) KummerM( - 3.14159265358979323846264338328, - 3.14159265358979323846264338328,1 ) = 2.71828182845904523536028747135 KummerU( - 3.14159265358979323846264338328, - 3.14159265358979323846264338328,1 ) = 19.2419564406028465661346237348 KummerM( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328 ) = 1.42892253084220157246185820464 KummerU( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328 ) = kummeru( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328) KummerM(26,1,0) = 1 KummerU(26,1,0) = kummeru(26,1,0) KummerM(26,3.14159265358979323846264338328,3.14159265358979323846264338328) = 3.91638029828702661403357541917e+5 KummerU(26,3.14159265358979323846264338328,3.14159265358979323846264338328) = 4.18995913372628050180833640475e-32 KummerM(26, - 3.14159265358979323846264338328,26) = 2.19471945265322419268333614674e+34 KummerU(26, - 3.14159265358979323846264338328,26) = 0 off rounded; clear x, y, z, t!*k!*c; df(df(KummerM(a,b,z),z),z); 2 (kummerm(2,3,z)*z - 2*kummerm(2,3,z)*z + 2*kummerm(2,3,z) + 2*kummerm(1,3,z)*z 2 - 4*kummerm(1,3,z) + 2*kummerm(0,3,z))/z df(KummerU(a,b,z),z); kummeru(2,3,z)*z - kummeru(2,3,z) - kummeru(1,3,z) ---------------------------------------------------- z z := ((x^2 + y)^5) + (x^(x+y)); x + y 10 8 6 2 4 3 2 4 5 z := x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y df(df(KummerM(a,b,z),y),x); 4*x + 4*y (x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 4*x + 4*y *log(x) *x + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4*x + 4*y *x + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3*x + 3*y *y + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 11 3*x + 3*y *log(x) *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 9 3*x + 3*y *log(x) *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 7 2 3*x + 3*y *log(x) *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 3 3*x + 3*y *log(x) *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3 4 3*x + 3*y *log(x) *x *y + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 3*x + 3*y *log(x) *x*y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3*x + 3*y *log(x) *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 3*x + 3*y *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 3*x + 3*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 3*x + 3*y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 3*x + 3*y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 3*x + 3*y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 2 3*x + 3*y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 3*x + 3*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2 3*x + 3*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 3*x + 3*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 3 3*x + 3*y *x *y + 60*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2 3*x + 3*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 3 3*x + 3*y *x *y + 30*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2 3*x + 3*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 4 3*x + 3*y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3 3*x + 3*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 4 3*x + 3*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 3 3*x + 3*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 5 3*x + 3*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 4 3*x + 3*y *x *y + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 3*x + 3*y *x*y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3*x + 3*y *x*y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3*x + 3*y *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 3*x + 3*y *y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3*x + 3*y *y + 5*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 5 3*x + 3*y *x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 3*x + 3*y 20*x x + y 10 8 6 2 4 3 2 4 5 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 3*x + 3*y 20*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 3*x + 3*y + 30*x x + y 10 8 6 2 4 3 2 4 5 5 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 3*x + 3*y + 30*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 3*x + 3*y + 20*x x + y 10 8 6 2 4 3 2 4 5 3 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 3*x + 3*y + 20*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 3*x + 3*y + 5*x x + y 10 8 6 2 4 3 2 4 5 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + 3*x + 3*y 5*x x + y 10 8 6 2 4 3 2 4 5 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + 3*x + 3*y x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + 2 3*x + 3*y *x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3*x + 3*y *log(x) *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3*x + 3*y *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 21 2*x + 2*y *log(x) *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 19 2*x + 2*y *log(x) *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 17 2 2*x + 2*y *log(x) *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 15 3 2*x + 2*y *log(x) *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 13 4 2*x + 2*y *log(x) *x *y + 252*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 11 5 2*x + 2*y *log(x) *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 9 6 2*x + 2*y *log(x) *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 7 7 2*x + 2*y *log(x) *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 8 2*x + 2*y *log(x) *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3 9 2*x + 2*y *log(x) *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 10 2*x + 2*y *log(x) *x*y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 2*x + 2*y *log(x) *x + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 21 2*x + 2*y *x + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 20 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 20 2*x + 2*y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 19 2*x + 2*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 19 2*x + 2*y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 18 2 2*x + 2*y *x *y + 180*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 18 2*x + 2*y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 17 2 2*x + 2*y *x *y + 90*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 17 2*x + 2*y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 16 3 2*x + 2*y *x *y + 720*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 16 2 2*x + 2*y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 15 3 2*x + 2*y *x *y + 360*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 15 2 2*x + 2*y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 4 2*x + 2*y *x *y + 1680*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 3 2*x + 2*y *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 13 4 2*x + 2*y *x *y + 840*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 13 3 2*x + 2*y *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 12 5 2*x + 2*y *x *y + 2520*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 12 4 2*x + 2*y *x *y + 252*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 5 2*x + 2*y *x *y + 1260*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 4 2*x + 2*y *x *y + 252*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 6 2*x + 2*y *x *y + 2520*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 5 2*x + 2*y *x *y - 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 2*x + 2*y *x + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 6 2*x + 2*y *x *y + 1260*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 5 2*x + 2*y *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 2*x + 2*y *x + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 7 2*x + 2*y *x *y + 1680*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 6 2*x + 2*y *x *y - 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 2*x + 2*y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 7 2*x + 2*y *x *y + 840*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 6 2*x + 2*y *x *y - 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2*x + 2*y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 8 2*x + 2*y *x *y + 720*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 7 2*x + 2*y *x *y - 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2 2*x + 2*y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 8 2*x + 2*y *x *y + 360*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 7 2*x + 2*y *x *y - 60*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2 2*x + 2*y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 9 2*x + 2*y *x *y + 180*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 8 2*x + 2*y *x *y - 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3 2*x + 2*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 9 2*x + 2*y *x *y + 90*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 8 2*x + 2*y *x *y - 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 3 2*x + 2*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 10 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 9 2*x + 2*y *x *y - 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 4 2*x + 2*y *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 2*x + 2*y *x*y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 2*x + 2*y *x*y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 2*x + 2*y *x*y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *x + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 2*x + 2*y *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *y + 10*x x + y 10 8 6 2 4 3 2 4 5 19 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y 10*x x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 50*x x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y 90*x x + y 10 8 6 2 4 3 2 4 5 17 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 90*x x + y 10 8 6 2 4 3 2 4 5 16 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 400*x x + y 10 8 6 2 4 3 2 4 5 16 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 360*x x + y 10 8 6 2 4 3 2 4 5 15 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 360*x x + y 10 8 6 2 4 3 2 4 5 14 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 1400*x x + y 10 8 6 2 4 3 2 4 5 14 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 840*x x + y 10 8 6 2 4 3 2 4 5 13 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 840*x x + y 10 8 6 2 4 3 2 4 5 12 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 2800*x x + y 10 8 6 2 4 3 2 4 5 12 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 1260*x x + y 10 8 6 2 4 3 2 4 5 11 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 1260*x x + y 10 8 6 2 4 3 2 4 5 10 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 3500*x x + y 10 8 6 2 4 3 2 4 5 10 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 2*x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y 1260*x x + y 10 8 6 2 4 3 2 4 5 9 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y 1260*x x + y 10 8 6 2 4 3 2 4 5 8 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 2800*x x + y 10 8 6 2 4 3 2 4 5 8 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 40*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y 840*x x + y 10 8 6 2 4 3 2 4 5 7 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 40*x x + y 10 8 6 2 4 3 2 4 5 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2*x + 2*y 840*x x + y 10 8 6 2 4 3 2 4 5 6 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 1400*x x + y 10 8 6 2 4 3 2 4 5 6 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 20*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 120*x x + y 10 8 6 2 4 3 2 4 5 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2*x + 2*y 360*x x + y 10 8 6 2 4 3 2 4 5 5 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 60*x x + y 10 8 6 2 4 3 2 4 5 5 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 360*x x + y 10 8 6 2 4 3 2 4 5 4 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 400*x x + y 10 8 6 2 4 3 2 4 5 4 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 40*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 120*x x + y 10 8 6 2 4 3 2 4 5 4 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 90*x x + y 10 8 6 2 4 3 2 4 5 3 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 40*x x + y 10 8 6 2 4 3 2 4 5 3 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 90*x x + y 10 8 6 2 4 3 2 4 5 2 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 50*x x + y 10 8 6 2 4 3 2 4 5 2 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y - 30*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 40*x x + y 10 8 6 2 4 3 2 4 5 2 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y - 2*x + 2*y 10*x x + y 10 8 6 2 4 3 2 4 5 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + 2*x + 2*y 10*x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y - 2*x + 2*y 8*x x + y 10 8 6 2 4 3 2 4 5 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y - 2*x + 2*y x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + 2 2*x + 2*y *x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 11 2*x + 2*y *log(x) *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 9 2*x + 2*y *log(x) *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 7 2 2*x + 2*y *log(x) *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 3 2*x + 2*y *log(x) *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3 4 2*x + 2*y *log(x) *x *y + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 2*x + 2*y *log(x) *x*y - 3*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 2*x + 2*y *log(x) *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 2*x + 2*y *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 2*x + 2*y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 2*x + 2*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 2*x + 2*y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 2 2*x + 2*y *x *y + 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2 2*x + 2*y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 3 2*x + 2*y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 3 2*x + 2*y *x *y + 60*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 4 2*x + 2*y *x *y + 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3 2*x + 2*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 4 2*x + 2*y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 3 2*x + 2*y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 5 2*x + 2*y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 4 2*x + 2*y *x *y + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2*x + 2*y *x*y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 2*x + 2*y *x*y - 3*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2*x + 2*y *y - 3*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *y + 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y 10*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2*x + 2*y 40*x x + y 10 8 6 2 4 3 2 4 5 7 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2*x + 2*y 40*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 60*x x + y 10 8 6 2 4 3 2 4 5 5 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 60*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 40*x x + y 10 8 6 2 4 3 2 4 5 3 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 40*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y 2*x + 2*y + 10*x x + y 10 8 6 2 4 3 2 4 5 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + 2*x + 2*y 10*x x + y 10 8 6 2 4 3 2 4 5 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + 2*x + 2*y x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + 2 2*x + 2*y *x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 2*x + 2*y *log(x) *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *x + 2*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) x + y *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 21 x + y *log(x) *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 19 x + y *log(x) *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 17 2 x + y *log(x) *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 15 3 x + y *log(x) *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 13 4 x + y *log(x) *x *y + 252*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 11 5 x + y *log(x) *x *y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 11 x + y *log(x) *x + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 9 6 x + y *log(x) *x *y - 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 9 x + y *log(x) *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 7 7 x + y *log(x) *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 7 2 x + y *log(x) *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 8 x + y *log(x) *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 3 x + y *log(x) *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3 9 x + y *log(x) *x *y - 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3 4 x + y *log(x) *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 10 x + y *log(x) *x*y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 x + y *log(x) *x*y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 30 x + y *x + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 29 x + y *x + 140*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 28 x + y *x *y + 70*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 27 x + y *x *y + 910*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 26 2 x + y *x *y + 455*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 25 2 x + y *x *y + 3640*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 24 3 x + y *x *y + 1820*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 23 3 x + y *x *y + 10010*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 22 4 x + y *x *y + 5005*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 21 4 x + y *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 21 x + y *x + 20020*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 20 5 x + y *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 20 x + y *x *y - 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 20 x + y *x + 10010*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 19 5 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 19 x + y *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 19 x + y *x + 30030*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 18 6 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 18 2 x + y *x *y - 180*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 18 x + y *x *y + 15015*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 17 6 x + y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 17 2 x + y *x *y - 90*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 17 x + y *x *y + 34320*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 16 7 x + y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 16 3 x + y *x *y - 720*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 16 2 x + y *x *y + 17160*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 15 7 x + y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 15 3 x + y *x *y - 360*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 15 2 x + y *x *y + 30030*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 8 x + y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 4 x + y *x *y - 1680*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 3 x + y *x *y + 15015*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 13 8 x + y *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 13 4 x + y *x *y - 840*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 13 3 x + y *x *y + 20020*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 12 9 x + y *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 12 5 x + y *x *y - 2520*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 12 4 x + y *x *y + 10010*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 9 x + y *x *y + 252*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 5 x + y *x *y - 1260*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 4 x + y *x *y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 x + y *x + 10010*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 10 x + y *x *y + 252*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 6 x + y *x *y - 2520*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 5 x + y *x *y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 x + y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 x + y *x + 5005*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 10 x + y *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 6 x + y *x *y - 1260*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 5 x + y *x *y - 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x + 3640*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 11 x + y *x *y + 210*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 7 x + y *x *y - 1680*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 6 x + y *x *y - 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 2 x + y *x *y + 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 x + y *x *y + 1820*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 11 x + y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 7 x + y *x *y - 840*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 6 x + y *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2 x + y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 x + y *x *y + 910*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 12 x + y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 8 x + y *x *y - 720*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 7 x + y *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 3 x + y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2 x + y *x *y + 455*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 12 x + y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 8 x + y *x *y - 360*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 7 x + y *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 3 x + y *x *y + 60*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2 x + y *x *y + 140*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 13 x + y *x *y + 45*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 9 x + y *x *y - 180*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 8 x + y *x *y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 4 x + y *x *y + 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3 x + y *x *y + 70*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 13 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 9 x + y *x *y - 90*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 8 x + y *x *y - 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 4 x + y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 3 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 14 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 10 x + y *x *y - 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 9 x + y *x *y - 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 5 x + y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 4 x + y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 x + y *x*y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 x + y *x*y - 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x*y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 x + y *x*y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 x + y *x*y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 x + y *y - x x + y 10 8 6 2 4 3 2 4 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 x + y *y + 5*x x + y 10 8 6 2 4 3 2 4 5 29 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 5*x x + y 10 8 6 2 4 3 2 4 5 28 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 100*x x + y 10 8 6 2 4 3 2 4 5 28 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 70*x x + y 10 8 6 2 4 3 2 4 5 27 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 70*x x + y 10 8 6 2 4 3 2 4 5 26 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 1300*x x + y 10 8 6 2 4 3 2 4 5 26 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 455*x x + y 10 8 6 2 4 3 2 4 5 25 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 455*x x + y 10 8 6 2 4 3 2 4 5 24 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 7800*x x + y 10 8 6 2 4 3 2 4 5 24 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 1820*x x + y 10 8 6 2 4 3 2 4 5 23 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 1820*x x + y 10 8 6 2 4 3 2 4 5 22 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 28600*x x + y 10 8 6 2 4 3 2 4 5 22 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5005*x x + y 10 8 6 2 4 3 2 4 5 21 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5005*x x + y 10 8 6 2 4 3 2 4 5 20 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 71500*x x + y 10 8 6 2 4 3 2 4 5 20 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + x x + y 10 8 6 2 4 3 2 4 5 20 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 10010*x x + y 10 8 6 2 4 3 2 4 5 19 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 10*x x + y 10 8 6 2 4 3 2 4 5 19 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 10010*x x + y 10 8 6 2 4 3 2 4 5 18 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 128700*x x + y 10 8 6 2 4 3 2 4 5 18 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 20*x x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 15015*x x + y 10 8 6 2 4 3 2 4 5 17 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 90*x x + y 10 8 6 2 4 3 2 4 5 17 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 15015*x x + y 10 8 6 2 4 3 2 4 5 16 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 171600*x x + y 10 8 6 2 4 3 2 4 5 16 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 45*x x + y 10 8 6 2 4 3 2 4 5 16 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 160*x x + y 10 8 6 2 4 3 2 4 5 16 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 17160*x x + y 10 8 6 2 4 3 2 4 5 15 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 360*x x + y 10 8 6 2 4 3 2 4 5 15 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 17160*x x + y 10 8 6 2 4 3 2 4 5 14 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 171600*x x + y 10 8 6 2 4 3 2 4 5 14 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 240*x x + y 10 8 6 2 4 3 2 4 5 14 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 560*x x + y 10 8 6 2 4 3 2 4 5 14 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 15015*x x + y 10 8 6 2 4 3 2 4 5 13 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 840*x x + y 10 8 6 2 4 3 2 4 5 13 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 15015*x x + y 10 8 6 2 4 3 2 4 5 12 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 128700*x x + y 10 8 6 2 4 3 2 4 5 12 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 630*x x + y 10 8 6 2 4 3 2 4 5 12 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1120*x x + y 10 8 6 2 4 3 2 4 5 12 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10010*x x + y 10 8 6 2 4 3 2 4 5 11 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1260*x x + y 10 8 6 2 4 3 2 4 5 11 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10010*x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 10 x + y *y + 71500*x x + y 10 8 6 2 4 3 2 4 5 10 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1008*x x + y 10 8 6 2 4 3 2 4 5 10 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1400*x x + y 10 8 6 2 4 3 2 4 5 10 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 5005*x x + y 10 8 6 2 4 3 2 4 5 9 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1260*x x + y 10 8 6 2 4 3 2 4 5 9 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 5005*x x + y 10 8 6 2 4 3 2 4 5 8 11 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 28600*x x + y 10 8 6 2 4 3 2 4 5 8 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1050*x x + y 10 8 6 2 4 3 2 4 5 8 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 1120*x x + y 10 8 6 2 4 3 2 4 5 8 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - x + y 40*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 1820*x x + y 10 8 6 2 4 3 2 4 5 7 11 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 840*x x + y 10 8 6 2 4 3 2 4 5 7 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 1820*x x + y 10 8 6 2 4 3 2 4 5 6 12 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 7800*x x + y 10 8 6 2 4 3 2 4 5 6 11 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 720*x x + y 10 8 6 2 4 3 2 4 5 6 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 560*x x + y 10 8 6 2 4 3 2 4 5 6 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 30*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 120*x x + y 10 8 6 2 4 3 2 4 5 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 455*x x + y 10 8 6 2 4 3 2 4 5 5 12 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 360*x x + y 10 8 6 2 4 3 2 4 5 5 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 60*x x + y 10 8 6 2 4 3 2 4 5 5 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 455*x x + y 10 8 6 2 4 3 2 4 5 4 13 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 1300*x x + y 10 8 6 2 4 3 2 4 5 4 12 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 315*x x + y 10 8 6 2 4 3 2 4 5 4 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 160*x x + y 10 8 6 2 4 3 2 4 5 4 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 50*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 120*x x + y 10 8 6 2 4 3 2 4 5 4 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 70*x x + y 10 8 6 2 4 3 2 4 5 3 13 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 90*x x + y 10 8 6 2 4 3 2 4 5 3 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 3 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 70*x x + y 10 8 6 2 4 3 2 4 5 2 14 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 100*x x + y 10 8 6 2 4 3 2 4 5 2 13 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 80*x x + y 10 8 6 2 4 3 2 4 5 2 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 20*x x + y 10 8 6 2 4 3 2 4 5 2 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 35*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 40*x x + y 10 8 6 2 4 3 2 4 5 2 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5*x x + y 10 8 6 2 4 3 2 4 5 14 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y x + y - 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + x + y 10*x x + y 10 8 6 2 4 3 2 4 5 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + x + y 5*x x + y 10 8 6 2 4 3 2 4 5 15 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y - x + y 9*x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + x + y 9*x x + y 10 8 6 2 4 3 2 4 5 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + x + y x + y 10 8 6 2 4 3 2 4 5 x *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 11 x + y *log(x) *x + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 9 x + y *log(x) *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 7 2 x + y *log(x) *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 3 x + y *log(x) *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 3 4 x + y *log(x) *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 5 x + y *log(x) *x*y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 20 x + y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 19 x + y *x + 180*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 18 x + y *x *y + 90*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 17 x + y *x *y + 720*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 16 2 x + y *x *y + 360*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 15 2 x + y *x *y + 1680*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 14 3 x + y *x *y + 840*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 13 3 x + y *x *y + 2520*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 12 4 x + y *x *y + 1260*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 4 x + y *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 11 x + y *x + 2520*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 5 x + y *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 x + y *x *y - 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 x + y *x + 1260*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 5 x + y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x *y - 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x + 1680*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 6 x + y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 2 x + y *x *y - 160*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 x + y *x *y + 840*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 6 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2 x + y *x *y - 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 x + y *x *y + 720*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 7 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 3 x + y *x *y - 240*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2 x + y *x *y + 360*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 7 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 3 x + y *x *y - 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2 x + y *x *y + 180*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 8 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 4 x + y *x *y - 160*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3 x + y *x *y + 90*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 8 x + y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 4 x + y *x *y - 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 3 x + y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 9 x + y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 5 x + y *x *y - 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 4 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x*y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 x + y *x*y - 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 x + y *x*y + x x + y 10 8 6 2 4 3 2 4 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 x + y *y + 10*x x + y 10 8 6 2 4 3 2 4 5 19 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 10*x x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 100*x x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 90*x x + y 10 8 6 2 4 3 2 4 5 17 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 90*x x + y 10 8 6 2 4 3 2 4 5 16 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 800*x x + y 10 8 6 2 4 3 2 4 5 16 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 360*x x + y 10 8 6 2 4 3 2 4 5 15 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 360*x x + y 10 8 6 2 4 3 2 4 5 14 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 2800*x x + y 10 8 6 2 4 3 2 4 5 14 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 840*x x + y 10 8 6 2 4 3 2 4 5 13 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 840*x x + y 10 8 6 2 4 3 2 4 5 12 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5600*x x + y 10 8 6 2 4 3 2 4 5 12 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 1260*x x + y 10 8 6 2 4 3 2 4 5 11 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 1260*x x + y 10 8 6 2 4 3 2 4 5 10 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 7000*x x + y 10 8 6 2 4 3 2 4 5 10 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 1260*x x + y 10 8 6 2 4 3 2 4 5 9 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 20*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 1260*x x + y 10 8 6 2 4 3 2 4 5 8 6 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5600*x x + y 10 8 6 2 4 3 2 4 5 8 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 15*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 40*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 840*x x + y 10 8 6 2 4 3 2 4 5 7 6 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 80*x x + y 10 8 6 2 4 3 2 4 5 7 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 840*x x + y 10 8 6 2 4 3 2 4 5 6 7 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 2800*x x + y 10 8 6 2 4 3 2 4 5 6 6 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 70*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 120*x x + y 10 8 6 2 4 3 2 4 5 6 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 360*x x + y 10 8 6 2 4 3 2 4 5 5 7 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 120*x x + y 10 8 6 2 4 3 2 4 5 5 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 360*x x + y 10 8 6 2 4 3 2 4 5 4 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 800*x x + y 10 8 6 2 4 3 2 4 5 4 7 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 110*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 120*x x + y 10 8 6 2 4 3 2 4 5 4 2 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 90*x x + y 10 8 6 2 4 3 2 4 5 3 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 80*x x + y 10 8 6 2 4 3 2 4 5 3 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 90*x x + y 10 8 6 2 4 3 2 4 5 2 9 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 100*x x + y 10 8 6 2 4 3 2 4 5 2 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y - 75*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 2 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y - x + y 20*x x + y 10 8 6 2 4 3 2 4 5 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + x + y 10*x x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y - x + y 19*x x + y 10 8 6 2 4 3 2 4 5 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + x + y 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 10 x + y *x + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x + 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 8 x + y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 x + y *x *y + 120*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 6 2 x + y *x *y + 60*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 2 x + y *x *y + 80*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 3 x + y *x *y + 40*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 3 x + y *x *y + 20*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2 4 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 4 x + y *x*y + 10*x x + y 10 8 6 2 4 3 2 4 5 9 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 10*x x + y 10 8 6 2 4 3 2 4 5 8 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 40*x x + y 10 8 6 2 4 3 2 4 5 7 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 40*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 60*x x + y 10 8 6 2 4 3 2 4 5 5 2 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 60*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 3 3 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 4 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + x + y 10*x x + y 10 8 6 2 4 3 2 4 5 5 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + x + y 10 8 6 2 4 3 2 4 5 38 50*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x x + y 10 8 6 2 4 3 2 4 5 + 900*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 36 *x *y + 7650 x + y 10 8 6 2 4 3 2 4 5 34 2 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 40800 x + y 10 8 6 2 4 3 2 4 5 32 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 153000 x + y 10 8 6 2 4 3 2 4 5 30 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 428400 x + y 10 8 6 2 4 3 2 4 5 28 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 60*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 28 *x + 928200 x + y 10 8 6 2 4 3 2 4 5 26 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 780*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 26 *x *y + 1591200 x + y 10 8 6 2 4 3 2 4 5 24 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 4680*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 24 2 *x *y + 2187900 x + y 10 8 6 2 4 3 2 4 5 22 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - 17160 x + y 10 8 6 2 4 3 2 4 5 22 3 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2431000 x + y 10 8 6 2 4 3 2 4 5 20 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - 42900 x + y 10 8 6 2 4 3 2 4 5 20 4 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2187900 x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 10 *y - 77220 x + y 10 8 6 2 4 3 2 4 5 18 5 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 60*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 18 *x + 1591200 x + y 10 8 6 2 4 3 2 4 5 16 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 11 *y - 102960 x + y 10 8 6 2 4 3 2 4 5 16 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 480*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 16 *x *y + 928200 x + y 10 8 6 2 4 3 2 4 5 14 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 12 *y - 102960 x + y 10 8 6 2 4 3 2 4 5 14 7 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 1680*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 14 2 *x *y + 428400 x + y 10 8 6 2 4 3 2 4 5 12 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 13 *y - 77220 x + y 10 8 6 2 4 3 2 4 5 12 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 3360*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 12 3 *x *y + 153000 x + y 10 8 6 2 4 3 2 4 5 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 14 *y - 42900 x + y 10 8 6 2 4 3 2 4 5 10 9 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 4200*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 10 4 *x *y + 40800 x + y 10 8 6 2 4 3 2 4 5 8 15 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - 17160 x + y 10 8 6 2 4 3 2 4 5 8 10 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 3360*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 8 5 *x *y + 7650 x + y 10 8 6 2 4 3 2 4 5 6 16 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 4680*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 6 11 *x *y + 1680 x + y 10 8 6 2 4 3 2 4 5 6 6 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 900*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 4 17 *x *y - 780 x + y 10 8 6 2 4 3 2 4 5 4 12 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 480*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 4 7 *x *y + 50 x + y 10 8 6 2 4 3 2 4 5 2 18 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 60*kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 13 *x *y + 60 x + y 10 8 6 2 4 3 2 4 5 2 8 *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 100*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 28 *x + 1300 x + y 10 8 6 2 4 3 2 4 5 26 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 7800*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 24 2 *x *y + 28600 x + y 10 8 6 2 4 3 2 4 5 22 3 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 71500 x + y 10 8 6 2 4 3 2 4 5 20 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 128700 x + y 10 8 6 2 4 3 2 4 5 18 5 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 160*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 18 *x + 171600 x + y 10 8 6 2 4 3 2 4 5 16 6 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 1280*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 16 *x *y + 171600 x + y 10 8 6 2 4 3 2 4 5 14 7 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 4480*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 14 2 *x *y + 128700 x + y 10 8 6 2 4 3 2 4 5 12 8 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 8960*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 12 3 *x *y + 71500 x + y 10 8 6 2 4 3 2 4 5 10 9 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - 11200 x + y 10 8 6 2 4 3 2 4 5 10 4 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 28600 x + y 10 8 6 2 4 3 2 4 5 8 10 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 8960*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 8 5 *x *y + 7800 x + y 10 8 6 2 4 3 2 4 5 6 11 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 4480*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 6 6 *x *y + 1300 x + y 10 8 6 2 4 3 2 4 5 4 12 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 1280*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 4 7 *x *y + 100 x + y 10 8 6 2 4 3 2 4 5 2 13 *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 160*kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 2 8 *x *y + 100 x + y 10 8 6 2 4 3 2 4 5 18 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 10 8 6 2 4 3 2 4 5 16 800*kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 2800 x + y 10 8 6 2 4 3 2 4 5 14 2 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 5600*kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 12 3 *x *y + 7000 x + y 10 8 6 2 4 3 2 4 5 10 4 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 5600*kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 8 5 *x *y + 2800 x + y 10 8 6 2 4 3 2 4 5 6 6 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 800*kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 4 7 *x *y + 100 x + y 10 8 6 2 4 3 2 4 5 2 8 *kummerm(0,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y ) 2*x + 2*y x + y 10 x + y 8 x + y 6 2 /(x*(x + 2*x *x + 10*x *x *y + 20*x *x *y x + y 4 3 x + y 2 4 x + y 5 20 18 + 20*x *x *y + 10*x *x *y + 2*x *y + x + 10*x *y 16 2 14 3 12 4 10 5 8 6 6 7 + 45*x *y + 120*x *y + 210*x *y + 252*x *y + 210*x *y + 120*x *y 4 8 2 9 10 + 45*x *y + 10*x *y + y )) df(KummerU(a,b,z),x); 2*x + 2*y (x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 2*x + 2*y *x + x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 2*x + 2*y x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + x + y x + y 10 8 6 2 4 3 2 4 5 x *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 11 x + y *log(x)*x + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 9 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 7 2 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 3 x + y *x *y + 5*x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 3 4 x + y *x *y + x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) 5 x + y *x*y - x x + y 10 8 6 2 4 3 2 4 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) x + y *x + x x + y 10 8 6 2 4 3 2 4 5 11 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y x + y 10 8 6 2 4 3 2 4 5 x *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 10 x + y *x *y + 10*x x + y 10 8 6 2 4 3 2 4 5 10 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + x + y 5*x x + y 10 8 6 2 4 3 2 4 5 9 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 5*x x + y 10 8 6 2 4 3 2 4 5 8 2 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 8 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + x + y 10*x x + y 10 8 6 2 4 3 2 4 5 7 2 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 6 3 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 60*x x + y 10 8 6 2 4 3 2 4 5 6 2 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 5 3 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 4 4 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 40*x x + y 10 8 6 2 4 3 2 4 5 4 3 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5*x x + y 10 8 6 2 4 3 2 4 5 3 4 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 5*x x + y 10 8 6 2 4 3 2 4 5 2 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + 10*x x + y 10 8 6 2 4 3 2 4 5 2 4 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y + x x + y 10 8 6 2 4 3 2 4 5 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y - x + y x + y 10 8 6 2 4 3 2 4 5 x *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) x + y *x + x x + y 10 8 6 2 4 3 2 4 5 6 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y - x + y x + y 10 8 6 2 4 3 2 4 5 x *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) x + y *y - x x + y 10 8 6 2 4 3 2 4 5 *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) x + y *x - x x + y 10 8 6 2 4 3 2 4 5 *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - x + y x + y 10 8 6 2 4 3 2 4 5 x *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) x + y 10 8 6 2 4 3 2 4 5 *y + 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 20 *x + 90 x + y 10 8 6 2 4 3 2 4 5 18 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 360*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 16 2 *x *y + 840 x + y 10 8 6 2 4 3 2 4 5 14 3 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 + 1260*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 12 4 *x *y + 1260 x + y 10 8 6 2 4 3 2 4 5 10 5 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 10 *x + 840 x + y 10 8 6 2 4 3 2 4 5 8 6 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 8 - 40*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + 360 x + y 10 8 6 2 4 3 2 4 5 6 7 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 6 - 60*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 2 *y + 90 x + y 10 8 6 2 4 3 2 4 5 4 8 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 4 - 40*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 3 *y + 10 x + y 10 8 6 2 4 3 2 4 5 2 9 *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 2 - 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 4 *y - 10 x + y 10 8 6 2 4 3 2 4 5 10 *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - x + y 10 8 6 2 4 3 2 4 5 8 40*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 - 60*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) 6 2 *x *y - 40 x + y 10 8 6 2 4 3 2 4 5 4 3 *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y x + y 10 8 6 2 4 3 2 4 5 2 - 10*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x 4 x + y 10 8 6 2 4 3 2 4 5 *y )/(x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )) % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 9. Bessel functions % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % Lengthy test of the Bessel functions. This isn't even % remotely exhaustive of the special cases -- though a % real person with lots of time could no doubt come up % with a better lot of tests than this automated rubbish. % Again, compromise by only actually doing one in five or % nine. If you want a really thorough test, you can % easily change this to get it; but it'll take hours to % run. clear p, q; hankel1(p,q); hankel1(p,q) r := df(ws,q); hankel1(p - 1,q)*q - hankel1(p,q)*p r := ------------------------------------- q on complex; r; besselj(p - 1,q)*q - besselj(p,q)*p + i*bessely(p - 1,q)*q - i*bessely(p,q)*p ------------------------------------------------------------------------------- q p:=3/8; 3 p := --- 8 r; - 5 3*pi - 5 3*pi (8*i*besselj(------,q)*cos(------)*q + 8*besselj(------,q)*sin(------)*q 8 8 8 8 - 3 5 + 3*i*besselj(------,q) + 8*i*besselj(---,q)*q 8 8 3 3*pi 3 3*pi - 3*i*besselj(---,q)*cos(------) - 3*besselj(---,q)*sin(------))/(8 8 8 8 8 3*pi *sin(------)*q) 8 q := pi; q := pi r; - 5 3*pi - 5 3*pi (8*i*besselj(------,pi)*cos(------)*pi + 8*besselj(------,pi)*sin(------)*pi 8 8 8 8 - 3 5 + 3*i*besselj(------,pi) + 8*i*besselj(---,pi)*pi 8 8 3 3*pi 3 3*pi - 3*i*besselj(---,pi)*cos(------) - 3*besselj(---,pi)*sin(------))/(8 8 8 8 8 3*pi *sin(------)*pi) 8 on rounded; *** Domain mode complex changed to complex-rounded r; - 0.119366207318921501826662822529 *hankel1(0.375,3.14159265358979323846264338328) + hankel1( - 0.625,3.14159265358979323846264338328) off complex, rounded; *** Domain mode complex-rounded changed to rounded df(df(BesselJ(pp,qq)+rr * hankel1(pp*2,qq) * BesselY(pp-qq,qq),qq),qq); 2 (besselj(pp - 2,qq)*qq - 2*besselj(pp - 1,qq)*pp*qq + besselj(pp - 1,qq)*qq 2 + besselj(pp,qq)*pp + besselj(pp,qq)*pp 2 + bessely(pp - qq,qq)*hankel1(2*pp - 2,qq)*qq *rr - 4*bessely(pp - qq,qq)*hankel1(2*pp - 1,qq)*pp*qq*rr + bessely(pp - qq,qq)*hankel1(2*pp - 1,qq)*qq*rr 2 + 4*bessely(pp - qq,qq)*hankel1(2*pp,qq)*pp *rr + 2*bessely(pp - qq,qq)*hankel1(2*pp,qq)*pp*rr 2 + df(bessely(pp - qq,qq),qq,2)*hankel1(2*pp,qq)*qq *rr 2 + 2*df(bessely(pp - qq,qq),qq)*hankel1(2*pp - 1,qq)*qq *rr 2 - 4*df(bessely(pp - qq,qq),qq)*hankel1(2*pp,qq)*pp*qq*rr)/qq % Possible values for real args t!*bes!*vr := { 1, pi, -pi, 26 }$ % Possible values for real and imaginary parts of complex args t!*bes!*vc := { 0, 3, -41/2 }$ array s!*bes(4)$ s!*bes(1) := "BesselJ"$ s!*bes(2) := "BesselY"$ s!*bes(3) := "BesselI"$ s!*bes(4) := "BesselK"$ pre := 16; pre := 16 precision pre; 30 preord := 10**pre; preord := 10000000000000000 t!*b!*c := 3; t*b*c := 3 algebraic procedure do!*one!*bessel(s,n,z); (if s = 1 then BesselJ(n,z) else if s = 2 then BesselY(n,z) else if s = 3 then BesselI(n,z) else BesselK(n,z)); do*one*bessel algebraic procedure pr!*bessel(s,n,z,k); << if t!*b!*c = k then << on rounded; bes1 := do!*one!*bessel(s,n,z); precision(pre+5); bes2 := do!*one!*bessel(s,n,z); if bes1 neq 0 then disc := floor abs(100*(bes2-bes1)*preord/bes1) else disc := 0; precision pre; write s!*bes(s), "(", n, ",", z, ") = ", bes1; if not numberp disc then << precom := !*complex; on complex; disc := disc; if not precom then off complex >>; if disc neq 0 then write " (discrepancy ", disc, "% of one s.f.)"; if numberp disc and disc > 200 then << write "***** WARNING Significant Inaccuracy."; write " Lower precision result:"; write " ", bes1; write " Higher precision result:"; precision(pre+5); write " ", bes2; precision pre >>; off rounded; t!*b!*c := 0 >>; t!*b!*c := t!*b!*c + 1 >>; pr*bessel % About to begin Bessel test. We have a list of possible % values, and we test every Bessel, with every value on the % list as both order and argument. Every Bessel is computed % twice, to different precisions (differing by 3), and any % discrepancy is reported. The value reported is the diff- % erence between the two computed values, expressed as a % percentage of the unit of the least significant displayed % digit. A discrepancy between 100% and 200% means that the % last digit of the displayed value was found to differ at % higher precision; values greater than 200% are cause for % concern. An ideal discrepancy would be between about 1% % and 20%; if the value is found to be zero, no discrepancy % is reported. off msg; for s := 1:4 do << write(" ... Testing ", s!*bes(s), " for real domains ... "); for each n in t!*bes!*vr do for each z in t!*bes!*vr do pr!*bessel(s, n, z, 5) >>; ... Testing BesselJ for real domains ... BesselJ(1, - 3.141592653589793) = - 0.2846153431797528 BesselJ(3.141592653589793,26) = - 0.006989220174690161 (discrepancy 5% of one s.f.) BesselJ(26,1) = 3.660826744416803e-35 ... Testing BesselY for real domains ... BesselY(1,3.141592653589793) = 0.358872916776719 BesselY(3.141592653589793, - 3.141592653589793) = 6.283185307179586 0.1545613960392598*( - 1.570796326794897) - 4.829362563540275 -------------------------------------------------------------------------------- 3.141592653589793 ( - 1.570796326794897) BesselY( - 3.141592653589793,26) = - 0.1386083623177915 ... Testing BesselI for real domains ... BesselI(1,1) = 0.565159103992485 BesselI(3.141592653589793,3.141592653589793) = 1.011423335928613 BesselI( - 3.141592653589793, - 3.141592653589793) = - 0.8856101155917482 + 0.4221616153281286*i BesselI(26,26) = 68397.86776155122 ... Testing BesselK for real domains ... BesselK(3.141592653589793,1) = 9.025908765806763 BesselK( - 3.141592653589793,3.141592653589793) = 0.1107526602738113 (discrepancy 1% of one s.f.) BesselK(26, - 3.141592653589793) = besselk(26, - 3.141592653589793) on complex; for s := 1:3 do << write (" ... Testing ", s!*bes(s), " for complex domains ... "); for each nr in t!*bes!*vc do for each ni in t!*bes!*vc do for each zr in t!*bes!*vc do for each zi in t!*bes!*vc do pr!*bessel(s, nr+ni*i, zr+zi*i, 9) >>; ... Testing BesselJ for complex domains ... BesselJ(0, - 20.5 + 3.0*i) = 1.05389016561334 + 1.410918160335249*i BesselJ(3*i, - 20.5 + 3.0*i) = 0.01225787392170983 + 0.01066256817009466*i BesselJ( - 20.5*i, - 20.5 + 3.0*i) = - 6.607837931625446e+38 + 7.203284455482089e+38*i BesselJ(3, - 20.5 + 3.0*i) = 1.568613483726435 - 0.7011991107137573*i BesselJ(3 + 3*i, - 20.5 + 3.0*i) = 0.007904103001381543 - 0.006566520928092784*i BesselJ(3.0 - 20.5*i, - 20.5 + 3.0*i) = - 7.069920310202644e+37 - 1.753271554229047e+37*i BesselJ( - 20.5, - 20.5 + 3.0*i) = 0.1758742246345278 - 0.332739860634916*i BesselJ( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = 0.08815299110072903 - 0.1369698556512304*i BesselJ( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 5.364748129151297e+46 + 2.608178375230083e+47*i ... Testing BesselY for complex domains ... BesselY(0, - 20.5 + 3.0*i) = - 1.404746667469566 + 1.060048452645186*i BesselY(3*i, - 20.5 + 3.0*i) = 0.4973091982659732 + 0.7985114801567726*i BesselY( - 20.5*i, - 20.5 + 3.0*i) = - 7.203284455482089e+38 - 6.607837931625446e+38*i (discrepancy 10% of one s.f.) BesselY(3, - 20.5 + 3.0*i) = 0.6963128100601111 + 1.576222640523309*i BesselY(3 + 3*i, - 20.5 + 3.0*i) = - 1.117333163968302 + 0.9789575771194796*i BesselY(3.0 - 20.5*i, - 20.5 + 3.0*i) = 1.753271554229047e+37 - 7.069920310202644e+37*i (discrepancy 10% of one s.f.) BesselY( - 20.5, - 20.5 + 3.0*i) = 0.2353954565395826 + 0.144691313932682*i BesselY( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = - 0.1527215881543493 + 0.2371137974094512*i BesselY( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 2.608178375230083e+47 - 5.364748129151297e+46*i (discrepancy 9% of one s.f.) ... Testing BesselI for complex domains ... BesselI(0, - 20.5 + 3.0*i) = - 6.891185608459107e+7 - 1.506065792318474e+7*i BesselI(3*i, - 20.5 + 3.0*i) = - 6879.511500081906 - 1745.250262122384*i BesselI( - 20.5*i, - 20.5 + 3.0*i) = 4.756052726395977e+40 - 1.836844915663626e+40*i BesselI(3, - 20.5 + 3.0*i) = 5.56813934269752e+7 + 1.026348768124686e+7*i BesselI(3 + 3*i, - 20.5 + 3.0*i) = 5917.619873410601 - 1360.623977225443*i BesselI(3.0 - 20.5*i, - 20.5 + 3.0*i) = 1.125266362182384e+40 - 3.487618645712341e+39*i BesselI( - 20.5, - 20.5 + 3.0*i) = - 0.0005935206748136158 - 0.001045512507192035*i BesselI( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = 0.0001367212493519011 - 0.00008899048591346719*i BesselI( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 4.705541285798881e+45 + 6.968045629188714e+45*i off complex; on msg; write (" ..."); ... write ("Bessel test complete."); Bessel test complete. % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 10. Incomplete Gamma and Beta functions (regularized) % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= igamma(3,0); 0 igamma(1,1); e - 1 ------- e igamma(1,4); 4 e - 1 -------- 4 e on rounded; igamma(1,1); 0.6321205588285577 igamma(1,4); 0.9816843611112658 igamma(2,4); 0.9084218055563291 igamma(0.5,4); 0.9953206825613376 ibeta(1,1,1); 1 ibeta(1,2,1); 1 ibeta(1,4,1); 1 ibeta(2,4,1); 1 ibeta(2,4,0.5); 0.8125 ibeta(0.12,0.43,0.9); 0.9114011367359803 precision 50; 16 ibeta(0.12,0.43,0.9); 0.91140113673598029129193960252752697713018702246564 precision 20; 50 ibeta(0.12,0.43,0.9); 0.91140113673598029129 on complex; *** Domain mode rounded changed to complex-rounded ibeta(1+i,1,1.5*i); - 0.12299596333468401978 + 0.28653674999354031595*i off rounded,complex; *** Domain mode complex-rounded changed to complex ibeta(3,2,x); 3 x *( - 3*x + 4) precision reset; 20 % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 11. Exponential Integral, Sine and Cosine Integrals; % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= procedure do!*one!*int!:function(fn,n); write (fn," of ", n, " is ", if fn=!Ei then Ei(n) else if fn=!Si then Si(n) else if fn=!Ci then Ci(n) else if fn=!Shi then Shi(n) else if fn=!Chi then Chi(n) else if fn=!Fresnel_!S then Fresnel_S(n) else if fn=!Fresnel_!C then Fresnel_C(n) ); do*one*int:function % Ei(0), Ci(0), and Chi(0) are undefined Ei(0); Ei(0) Ci(0); Ci(0) Chi(0); Chi(0) on rounded; Ei(0); ***** Ei(0) is undefined Ci(0); ***** Ci(0) is undefined Chi(0); ***** Chi(0) is undefined % Chi is undefined for negative real argument Chi(-1); 3.14159265359*i + 0.83786694098 for x := 0.1 step 0.1 until 2.0 do do!*one!*int!:function(!Ei,x); Ei of 0.1 is - 1.62281281397 Ei of 0.2 is - 0.821760587902 Ei of 0.3 is - 0.302668539266 Ei of 0.4 is 0.104765218619 Ei of 0.5 is 0.454219904863 Ei of 0.6 is 0.769881289937 Ei of 0.7 is 1.06490719462 Ei of 0.8 is 1.34739654821 Ei of 0.9 is 1.6228117137 Ei of 1 is 1.89511781636 Ei of 1.1 is 2.16737827956 Ei of 1.2 is 2.44209228519 Ei of 1.3 is 2.72139888023 Ei of 1.4 is 3.00720746415 Ei of 1.5 is 3.30128544913 Ei of 1.6 is 3.60531994902 Ei of 1.7 is 3.92096320135 Ei of 1.8 is 4.24986755749 Ei of 1.9 is 4.59371368695 Ei of 2.0 is 4.954234356 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Ei,x); Ei of - 0.1 is - 1.82292395842 Ei of - 0.2 is - 1.22265054418 Ei of - 0.3 is - 0.905676651676 Ei of - 0.4 is - 0.702380118866 Ei of - 0.5 is - 0.559773594776 Ei of - 0.6 is - 0.454379503189 Ei of - 0.7 is - 0.373768843234 Ei of - 0.8 is - 0.310596578546 Ei of - 0.9 is - 0.260183939326 Ei of -1 is - 0.219383934396 Ei of - 1.1 is - 0.185990904536 Ei of - 1.2 is - 0.158408436851 Ei of - 1.3 is - 0.135450957849 Ei of - 1.4 is - 0.116219312571 Ei of - 1.5 is - 0.100019582407 Ei of - 1.6 is - 0.0863083336975 Ei of - 1.7 is - 0.0746546444013 Ei of - 1.8 is - 0.0647131293639 Ei of - 1.9 is - 0.0562043781745 Ei of - 2.0 is - 0.0489005107081 for x := 0.0 step 0.1 until 2.0 do do!*one!*int!:function(!Si,x); Si of 0 is 0 Si of 0.1 is 0.0999444611083 Si of 0.2 is 0.199556088526 Si of 0.3 is 0.298504043807 Si of 0.4 is 0.396461464751 Si of 0.5 is 0.493107418043 Si of 0.6 is 0.588128809608 Si of 0.7 is 0.681222239117 Si of 0.8 is 0.772095785482 Si of 0.9 is 0.860470710745 Si of 1 is 0.946083070367 Si of 1.1 is 1.02868521867 Si of 1.2 is 1.10804719901 Si of 1.3 is 1.18395800908 Si of 1.4 is 1.25622673278 Si of 1.5 is 1.32468353117 Si of 1.6 is 1.38918048587 Si of 1.7 is 1.44959228968 Si of 1.8 is 1.50581678026 Si of 1.9 is 1.55777531375 Si of 2.0 is 1.6054129768 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Si,x); Si of - 0.1 is - 0.0999444611083 Si of - 0.2 is - 0.199556088526 Si of - 0.3 is - 0.298504043807 Si of - 0.4 is - 0.396461464751 Si of - 0.5 is - 0.493107418043 Si of - 0.6 is - 0.588128809608 Si of - 0.7 is - 0.681222239117 Si of - 0.8 is - 0.772095785482 Si of - 0.9 is - 0.860470710745 Si of -1 is - 0.946083070367 Si of - 1.1 is - 1.02868521867 Si of - 1.2 is - 1.10804719901 Si of - 1.3 is - 1.18395800908 Si of - 1.4 is - 1.25622673278 Si of - 1.5 is - 1.32468353117 Si of - 1.6 is - 1.38918048587 Si of - 1.7 is - 1.44959228968 Si of - 1.8 is - 1.50581678026 Si of - 1.9 is - 1.55777531375 Si of - 2.0 is - 1.6054129768 for x := 0.1 step 0.1 until 2.0 do do!*one!*int!:function(!Ci,x); Ci of 0.1 is - 1.72786838666 Ci of 0.2 is - 1.04220559567 Ci of 0.3 is - 0.649172932971 Ci of 0.4 is - 0.378809346425 Ci of 0.5 is - 0.177784078807 Ci of 0.6 is - 0.0222707069593 Ci of 0.7 is 0.100514707009 Ci of 0.8 is 0.198278615952 Ci of 0.9 is 0.276067830468 Ci of 1 is 0.337403922901 Ci of 1.1 is 0.384873377425 Ci of 1.2 is 0.420459182894 Ci of 1.3 is 0.445738567529 Ci of 1.4 is 0.462006585095 Ci of 1.5 is 0.470356317195 Ci of 1.6 is 0.471732516932 Ci of 1.7 is 0.466968364177 Ci of 1.8 is 0.456811129418 Ci of 1.9 is 0.441940349682 Ci of 2.0 is 0.422980828775 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Ci,x); Ci of - 0.1 is 3.14159265359*i - 1.72786838666 Ci of - 0.2 is 3.14159265359*i - 1.04220559567 Ci of - 0.3 is 3.14159265359*i - 0.649172932971 Ci of - 0.4 is 3.14159265359*i - 0.378809346425 Ci of - 0.5 is 3.14159265359*i - 0.177784078807 Ci of - 0.6 is 3.14159265359*i - 0.0222707069593 Ci of - 0.7 is 3.14159265359*i + 0.100514707009 Ci of - 0.8 is 3.14159265359*i + 0.198278615952 Ci of - 0.9 is 3.14159265359*i + 0.276067830468 Ci of -1 is 3.14159265359*i + 0.337403922901 Ci of - 1.1 is 3.14159265359*i + 0.384873377425 Ci of - 1.2 is 3.14159265359*i + 0.420459182894 Ci of - 1.3 is 3.14159265359*i + 0.445738567529 Ci of - 1.4 is 3.14159265359*i + 0.462006585095 Ci of - 1.5 is 3.14159265359*i + 0.470356317195 Ci of - 1.6 is 3.14159265359*i + 0.471732516932 Ci of - 1.7 is 3.14159265359*i + 0.466968364177 Ci of - 1.8 is 3.14159265359*i + 0.456811129418 Ci of - 1.9 is 3.14159265359*i + 0.441940349682 Ci of - 2.0 is 3.14159265359*i + 0.422980828775 for x := 0.0 step 0.1 until 2.0 do do!*one!*int!:function(!Shi,x); Shi of 0 is 0 Shi of 0.1 is 0.100055572225 Shi of 0.2 is 0.200444978141 Shi of 0.3 is 0.301504056205 Shi of 0.4 is 0.403572668742 Shi of 0.5 is 0.50699674982 Shi of 0.6 is 0.612130396563 Shi of 0.7 is 0.719338018929 Shi of 0.8 is 0.828996563379 Shi of 0.9 is 0.941497826511 Shi of 1 is 1.05725087538 Shi of 1.1 is 1.17668459205 Shi of 1.2 is 1.30025036102 Shi of 1.3 is 1.42842491904 Shi of 1.4 is 1.56171338836 Shi of 1.5 is 1.70065251577 Shi of 1.6 is 1.84581414136 Shi of 1.7 is 1.99780892288 Shi of 1.8 is 2.15729034343 Shi of 1.9 is 2.32495903256 Shi of 2.0 is 2.50156743335 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Shi,x); Shi of - 0.1 is - 0.100055572225 Shi of - 0.2 is - 0.200444978141 Shi of - 0.3 is - 0.301504056205 Shi of - 0.4 is - 0.403572668742 Shi of - 0.5 is - 0.50699674982 Shi of - 0.6 is - 0.612130396563 Shi of - 0.7 is - 0.719338018929 Shi of - 0.8 is - 0.828996563379 Shi of - 0.9 is - 0.941497826511 Shi of -1 is - 1.05725087538 Shi of - 1.1 is - 1.17668459205 Shi of - 1.2 is - 1.30025036102 Shi of - 1.3 is - 1.42842491904 Shi of - 1.4 is - 1.56171338836 Shi of - 1.5 is - 1.70065251577 Shi of - 1.6 is - 1.84581414136 Shi of - 1.7 is - 1.99780892288 Shi of - 1.8 is - 2.15729034343 Shi of - 1.9 is - 2.32495903256 Shi of - 2.0 is - 2.50156743335 for x := 0.1 step 0.1 until 2.0 do do!*one!*int!:function(!Chi,x); Chi of 0.1 is - 1.72286838619 Chi of 0.2 is - 1.02220556604 Chi of 0.3 is - 0.604172595471 Chi of 0.4 is - 0.298807450123 Chi of 0.5 is - 0.0527768449565 Chi of 0.6 is 0.157750893374 Chi of 0.7 is 0.345569175695 Chi of 0.8 is 0.518399984833 Chi of 0.9 is 0.681313887185 Chi of 1 is 0.83786694098 Chi of 1.1 is 0.990693687514 Chi of 1.2 is 1.14184192417 Chi of 1.3 is 1.29297396119 Chi of 1.4 is 1.44549407579 Chi of 1.5 is 1.60063293336 Chi of 1.6 is 1.75950580766 Chi of 1.7 is 1.92315427848 Chi of 1.8 is 2.09257721406 Chi of 1.9 is 2.26875465439 Chi of 2.0 is 2.45266692265 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Chi,x); Chi of - 0.1 is 3.14159265359*i - 1.72286838619 Chi of - 0.2 is 3.14159265359*i - 1.02220556604 Chi of - 0.3 is 3.14159265359*i - 0.604172595471 Chi of - 0.4 is 3.14159265359*i - 0.298807450123 Chi of - 0.5 is 3.14159265359*i - 0.0527768449565 Chi of - 0.6 is 3.14159265359*i + 0.157750893374 Chi of - 0.7 is 3.14159265359*i + 0.345569175695 Chi of - 0.8 is 3.14159265359*i + 0.518399984833 Chi of - 0.9 is 3.14159265359*i + 0.681313887185 Chi of -1 is 3.14159265359*i + 0.83786694098 Chi of - 1.1 is 3.14159265359*i + 0.990693687514 Chi of - 1.2 is 3.14159265359*i + 1.14184192417 Chi of - 1.3 is 3.14159265359*i + 1.29297396119 Chi of - 1.4 is 3.14159265359*i + 1.44549407579 Chi of - 1.5 is 3.14159265359*i + 1.60063293336 Chi of - 1.6 is 3.14159265359*i + 1.75950580766 Chi of - 1.7 is 3.14159265359*i + 1.92315427848 Chi of - 1.8 is 3.14159265359*i + 2.09257721406 Chi of - 1.9 is 3.14159265359*i + 2.26875465439 Chi of - 2.0 is 3.14159265359*i + 2.45266692265 for x := 0.1 step 0.1 until 2.0 do do!*one!*int!:function(!Fresnel_!S,x); Fresnel_S of 0.1 is 0.000523589547612 Fresnel_S of 0.2 is 0.00418760916166 Fresnel_S of 0.3 is 0.0141169980066 Fresnel_S of 0.4 is 0.0333594326606 Fresnel_S of 0.5 is 0.06473243286 Fresnel_S of 0.6 is 0.110540207359 Fresnel_S of 0.7 is 0.172136457863 Fresnel_S of 0.8 is 0.249341393054 Fresnel_S of 0.9 is 0.339776344393 Fresnel_S of 1 is 0.43825914739 Fresnel_S of 1.1 is 0.536497911097 Fresnel_S of 1.2 is 0.623400918546 Fresnel_S of 1.3 is 0.686333285535 Fresnel_S of 1.4 is 0.713525077363 Fresnel_S of 1.5 is 0.697504960082 Fresnel_S of 1.6 is 0.638887683509 Fresnel_S of 1.7 is 0.549195940322 Fresnel_S of 1.8 is 0.450938769268 Fresnel_S of 1.9 is 0.373347317817 Fresnel_S of 2.0 is 0.343415678364 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Fresnel_!S,x); Fresnel_S of - 0.1 is - 0.000523589547612 Fresnel_S of - 0.2 is - 0.00418760916166 Fresnel_S of - 0.3 is - 0.0141169980066 Fresnel_S of - 0.4 is - 0.0333594326606 Fresnel_S of - 0.5 is - 0.06473243286 Fresnel_S of - 0.6 is - 0.110540207359 Fresnel_S of - 0.7 is - 0.172136457863 Fresnel_S of - 0.8 is - 0.249341393054 Fresnel_S of - 0.9 is - 0.339776344393 Fresnel_S of -1 is - 0.43825914739 Fresnel_S of - 1.1 is - 0.536497911097 Fresnel_S of - 1.2 is - 0.623400918546 Fresnel_S of - 1.3 is - 0.686333285535 Fresnel_S of - 1.4 is - 0.713525077363 Fresnel_S of - 1.5 is - 0.697504960082 Fresnel_S of - 1.6 is - 0.638887683509 Fresnel_S of - 1.7 is - 0.549195940322 Fresnel_S of - 1.8 is - 0.450938769268 Fresnel_S of - 1.9 is - 0.373347317817 Fresnel_S of - 2.0 is - 0.343415678364 for x := 0.1 step 0.1 until 2.0 do do!*one!*int!:function(!Fresnel_!C,x); Fresnel_C of 0.1 is 0.0999975326271 Fresnel_C of 0.2 is 0.199921057594 Fresnel_C of 0.3 is 0.299400976052 Fresnel_C of 0.4 is 0.397480759172 Fresnel_C of 0.5 is 0.492344225871 Fresnel_C of 0.6 is 0.581095446992 Fresnel_C of 0.7 is 0.659652351905 Fresnel_C of 0.8 is 0.722844171896 Fresnel_C of 0.9 is 0.764823021273 Fresnel_C of 1 is 0.779893400377 Fresnel_C of 1.1 is 0.763806666062 Fresnel_C of 1.2 is 0.715437722923 Fresnel_C of 1.3 is 0.638550454727 Fresnel_C of 1.4 is 0.543095783546 Fresnel_C of 1.5 is 0.44526117604 Fresnel_C of 1.6 is 0.36546168344 Fresnel_C of 1.7 is 0.323826876004 Fresnel_C of 1.8 is 0.333632927222 Fresnel_C of 1.9 is 0.394470534892 Fresnel_C of 2.0 is 0.488253406075 for x := -0.1 step -0.1 until -2.0 do do!*one!*int!:function(!Fresnel_!C,x); Fresnel_C of - 0.1 is - 0.0999975326271 Fresnel_C of - 0.2 is - 0.199921057594 Fresnel_C of - 0.3 is - 0.299400976052 Fresnel_C of - 0.4 is - 0.397480759172 Fresnel_C of - 0.5 is - 0.492344225871 Fresnel_C of - 0.6 is - 0.581095446992 Fresnel_C of - 0.7 is - 0.659652351905 Fresnel_C of - 0.8 is - 0.722844171896 Fresnel_C of - 0.9 is - 0.764823021273 Fresnel_C of -1 is - 0.779893400377 Fresnel_C of - 1.1 is - 0.763806666062 Fresnel_C of - 1.2 is - 0.715437722923 Fresnel_C of - 1.3 is - 0.638550454727 Fresnel_C of - 1.4 is - 0.543095783546 Fresnel_C of - 1.5 is - 0.44526117604 Fresnel_C of - 1.6 is - 0.36546168344 Fresnel_C of - 1.7 is - 0.323826876004 Fresnel_C of - 1.8 is - 0.333632927222 Fresnel_C of - 1.9 is - 0.394470534892 Fresnel_C of - 2.0 is - 0.488253406075 off rounded; % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 12. Dilogarithm, polylogartihm and Lerch_phi % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= polylog(n,0); 0 polylog(2,1); 2 pi ----- 6 polylog(3,1); zeta(3) polylog(2,i); 2 48*catalan*i - pi -------------------- 48 df(polylog(a,x),x); - log( - x + 1) ------------------ x polylog(1,x); - log( - x + 1) precision reset; 12 on rounded; polylog(2,1/3); 0.366213229977 off rounded; Lerch_phi(3,4,1); polylog(4,3) -------------- 3 Lerch_phi(4,0,3); - 1 ------ 3 Lerch_phi(x,a,1); polylog(2,x) -------------- x Lerch_phi(1,x,1); zeta(x) df(Lerch_phi(x,3,4),x); - 4*lerch_phi(x,3,4) + lerch_phi(x,2,4) ------------------------------------------ x % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= % 13. Constants % =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= on rounded; off complex; precision 50; 12 Euler_gamma; 0.57721566490153286060651209008240243104215933593992 Khinchin; 2.6854520010653064453097148354817956938203822939945 golden_ratio; 1.6180339887498948482045868343656381177203091798058 Catalan; 0.91596559417721901505460351493238411077414937428167 on complex; *** Domain mode rounded changed to complex-rounded Euler_gamma; 0.57721566490153286060651209008240243104215933593992 Khinchin; 2.6854520010653064453097148354817956938203822939945 golden_ratio; 1.6180339887498948482045868343656381177203091798058 Catalan; 0.91596559417721901505460351493238411077414937428167 end;