Function: intmellininv
Section: sums
C-Name: intmellininv0
Prototype: V=GGEDGp
Help: intmellininv(X=sig,z,expr,{tab}): numerical integration on the
 line real(X) = sig (or sig[1]) of expr(X)z^(-X)dX/(2*I*Pi), i.e. inverse Mellin
 transform of s at x. sig is coded as follows: either it is real, and then
 by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is
 the abscissa of integration, and al = 0 for slowly decreasing functions, or
 al > 0 if s(z) decreases like exp(-al*z). tab is as in intnum. Use
 intmellininvshort if several values must be computed.
Wrapper: (,,G)
Description:
  (gen,gen,gen,?gen):gen:prec intmellininv(${3 cookie}, ${3 wrapper}, $1, $2, $4, prec)
Doc: numerical
 integration of $(2i\pi)^{-1}\var{expr}(X)z^{-X}$ with respect to $X$ on the
 line $\Re(X)=sig$,  in other words, inverse Mellin transform of
 the function corresponding to \var{expr} at the value $z$.

 $sig$ is coded as follows. Either it is a real number $\sigma$, equal to the
 abscissa of integration, and then the integrated is assumed to decrease
 exponentially fast, of the order of $\exp(-t)$ when the imaginary part of the
 variable tends to $\pm\infty$. Or it is a two component vector
 $[\sigma,\alpha]$, where $\sigma$ is as before, and either $\alpha=0$ for
 slowly decreasing functions, or $\alpha>0$ for functions decreasing like
 $\exp(-\alpha t)$, such as gamma products. Note that it is not necessary to
 choose the exact value of $\alpha$, and that $\alpha=1$ (equivalent to $sig$
 alone) is usually sufficient. \var{tab} is as in \kbd{intnum}.

 As all similar functions, this function is provided for the convenience of
 the user, who could use \kbd{intnum} directly. However it is in general
 better to use \kbd{intmellininvshort}.

 \bprog
 ? \p 105
 ? intmellininv(s=2,4, gamma(s)^3);
 time = 1,190 ms. \\@com reasonable.
 ? \p 308
 ? intmellininv(s=2,4, gamma(s)^3);
 time = 51,300 ms. \\@com slow because of $\Gamma(s)^3$.
 @eprog\noindent

 \synt{intmellininv}{void *E, GEN (*eval)(void*,GEN), GEN sig, GEN z, GEN tab, long prec}.
