HP71B Integral Questions

[Albert Chan]

Out of curiosity, I tried using the actual Gamma function integral

\(\Gamma(x+1) = \int _0 ^1 (-\ln(t))^x dt \)

This integral, for any big x, choke on HP71B INTEGRAL

Instead, I tried letting \(t = s^{x+1}\), turning the extreme spike into a bell-shaped curve

\(\Gamma(x+1) = \int _0^1 (x+1) s^x (-(x+1)\ln(s))^x ds = (x+1)^{x+1} \int _0^1 (-s\ln(s))^x ds\)

Scale RHS 2 terms to avoid hitting overflow limits, we have:

Code:

10 P=1/10^8
20 DEF FNA(X)=INTEGRAL(0,1,P,(-LN(IVAR))^X)
30 DEF FNB(X)=100*((X+1)/100)^(X+1)*INTEGRAL(0,1,P,(-100*IVAR*LN(IVAR))^X)
40 INPUT "X=?";X
50 T=TIME @ DISP FNA(X),IBOUND,TIME-T
60 T=TIME @ DISP FNB(X),IBOUND,TIME-T
70 DISP GAMMA(X+1),"GAMMA(X+1)"

>RUN
X=?10
3623769.01194     -3.61915267163E-2     40.36
3628800.00001     12719498.961          .09
3628800           GAMMA(X+1)
>RUN
X=?20
1.66431877492E18  -15440997253.2        40.47
2.4329020082E18   4.16433765069E22      .11
2.43290200818E18  GAMMA(X+1)
>RUN
X=?30
1.39229859877E31  -1.13405116532E23     40.31
2.65252859812E32  1.55401137325E38      .19
2.65252859812E32  GAMMA(X+1)