1asympexp.mac is from "The Use of Symbolic Computation in Perturbation 2Analysis" by R. H. Rand in Symbolic Computation in Fluid Mechanics and 3Heat Transfer ed H.H.Bau (ASME 1988) (http://tam.cornell.edu/Rand.html) 4 5The routine approximates definite integrals of the form 6 7 b 8 / 9 | x phi(t) 10 | f(t) %e dt 11 | 12 / 13 a 14 15in the limit as x approaches infinity. 16 17The idea of the method is that exp(x*phi(t)) makes its largest 18contribution to the integral in the neighbourhood of the point t=c at 19which phi(t) is maximum. 20 21Example 1. 22---------- 23 24The modified Bessel function of the first kind can be expressed as 25 26 27 %pi 28 / 29 1 | x cos(t) 30 I (x) = - | cos(n*t) %e dt 31 n %pi | 32 / 33 0 34 35The results below from maxima-5.9.0-cvs match those in the paper. 36 37(C1) load("./asympexp.mac"); 38(D1) ./asympexp.mac 39(C2) asymptotic(); 40The integrand is of the form: f(t) exp(x phi(t)) 41enter f(t) 42cos(n*t); 43enter phi(t) 44cos(t); 45enter the lower limit of integration 460; 47enter the upper limit of integration 48%pi; 49 COS(t) x 50The integrand is COS(n t) %E 51integrated from 0 to %PI 52enter value of t at which phi = COS(t) is maximum 530; 54enter truncation order 554; 56 4 2 3 3 4 2 57(D2) SQRT(2) SQRT(%PI) (98304 x - 49152 n x + 12288 x + 12288 n x 58 59 2 2 2 6 4 2 60 - 30720 n x + 6912 x - 2048 n x + 17920 n x - 33152 n x + 7200 x 61 62 8 6 4 2 x 9/2 63 + 256 n - 5376 n + 31584 n - 51664 n + 11025) %E /(196608 x ) 64(C3) time(d2); 65Time: 66(D3) [2.223] 67 68 69Example 2. 70---------- 71 72The second example from the paper is Stirling's formula for the gamma function 73 74 infinity 75 / 76 | 1 -t+x*log(t) 77 Gamma(x) = | - %e dt 78 | t 79 / 80 0 81 82or by setting u = t/x 83 84 infinity 85 / 86 | 1 x(log(u)-u) 87 Gamma(x) = x | - %e du 88 | u 89 / 90 0 91 92The results below from maxima-5.9.0-cvs match those in the paper. 93 94(C4) asymptotic(); 95The integrand is of the form: f(t) exp(x phi(t)) 96enter f(t) 971/t; 98enter phi(t) 99log(t)-t; 100enter the lower limit of integration 1010; 102enter the upper limit of integration 103inf; 104 (LOG(t) - t) x 105 %E 106The integrand is ---------------- 107 t 108integrated from 0 to INF 109enter value of t at which phi = LOG(t) - t is maximum 1101; 111enter truncation order 1124; 113 4 3 2 - x 114 SQRT(2) SQRT(%PI) (2488320 x + 207360 x + 8640 x - 6672 x - 571) %E 115(D4) ------------------------------------------------------------------------- 116 9/2 117 2488320 x 118(C5) time(d4); 119Time: 120(D5) [10.045] 121 122 123Local Variables: *** 124mode: Text *** 125End: ***