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: ***