Function: intmellininvshort
Section: sums
C-Name: intmellininvshort
Prototype: GGGp
Help: intmellininvshort(sig,z,tab): numerical integration on the
 line real(X) = sig (or sig[1]) of s(X)z^(-X)dX/(2*I*Pi), i.e. inverse Mellin
 transform of s at z. sig is coded as follows: either it is real, and then
 by default assume s(X) decreases like exp(-X). Or sig = [sigR, al], sigR is
 the abscissa of integration, and al = 0 for slowly decreasing functions, or
 al > 0 if s(X) decreases like exp(-al*X). Compulsory table tab has been
 precomputed using the command intfuncinit(t=[[-1],sig[2]],[[1],sig[2]],s)
 (with possibly its two optional additional parameters), where sig[2] = 1
 if not given. Orders of magnitude faster than intmellininv.
Doc: numerical integration
 of $(2i\pi)^{-1}s(X)z^{-X}$ with respect to $X$ on the line $\Re(X)=sig$.
 In other words, inverse Mellin transform of $s(X)$ at the value $z$.
 Here $s(X)$ is implicitly contained in \var{tab} in \kbd{intfuncinit} format,
 typically
 \bprog
 tab = intfuncinit(T = [-1], [1], s(sig + I*T))
 @eprog\noindent
 or similar commands. Take the example of the inverse Mellin transform of
 $\Gamma(s)^3$ given in \kbd{intmellininv}:

 \bprog
 ? \p 105
 ? oo = [1]; \\@com for clarity
 ? A = intmellininv(s=2,4, gamma(s)^3);
 time = 2,500 ms. \\@com not too fast because of $\Gamma(s)^3$.
 \\ @com function of real type, decreasing as $\exp(-3\pi/2\cdot |t|)$
 ? tab = intfuncinit(t=[-oo, 3*Pi/2],[oo, 3*Pi/2], gamma(2+I*t)^3, 1);
 time = 1,370 ms.
 ? intmellininvshort(2,4, tab) - A
 time = 50 ms.
 %4 = -1.26... - 3.25...E-109*I \\@com 50 times faster than \kbd{A} and perfect.
 ? tab2 = intfuncinit(t=-oo, oo, gamma(2+I*t)^3, 1);
 ? intmellininvshort(2,4, tab2)
 %6 = -1.2...E-42 - 3.2...E-109*I  \\@com 63 digits lost
 @eprog\noindent
 In the computation of \var{tab}, it was not essential to include the
 \emph{exact} exponential decrease of $\Gamma(2+it)^3$. But as the last
 example shows, a rough indication \emph{must} be given, otherwise slow
 decrease is assumed, resulting in catastrophic loss of accuracy.
