I have been attempting to investigate the function
f(v) = intnumgauss(x=0,1, log(abs(x^v - (1 - x)^v))/x)
for various positive values of v.
I cannot use the standard intnum because I get a domain error for log.
Even running on ~1000 precision I can't seem to get stable values. For example:
? f(2)
%2 = -2.4225110046071715036046951871135247939
? \p 100
realprecision = 115 significant digits (100 digits displayed)
? f(2)
%3 = -2.448045467855465962637520950271434461626906750839030270690427142653795442332270916763571861586211446
? \p 500
realprecision = 500 significant digits
? f(2)
%4 = -2.46236623125534737591438595461875712342066622953365858291276735021494398975080267620885683763596204514917830504241706038254418887945208437689348318484881426720
6369967589343063020562790497652612860415817451002780032725265928653909391217140691580623706469465927763263097639596472178685417890225073853769466120537250499583029234
0105930878210069637749954513919387562716196350765008607089175830388519925771160765046638359503537136422685591455325755980613085912846141203505510512236119034080952188
143076548
I have several questions:
A. Why does intnum not work for this function? The similar integral g(v)=intnum(x=0,1,log(x^v+(1-x)^v)/x) is evaluated just fine, and agrees with intnumgauss.
B. How should I best compute f? Suppose I wanted many accurate digits for several special values, to try and recognize them.
C. Should I even be trusting intnumgauss for such an integrand?