gamma

Function: gamma
Section: transcendental
C-Name: ggamma
Prototype: Gp
Help: gamma(s): gamma function at s, a complex or p-adic number, or a series.
Doc: For $s$ a complex number, evaluates Euler's gamma
 function \sidx{gamma-function}
 $$\Gamma(s)=\int_0^\infty t^{s-1}\exp(-t)\,dt.$$
 Error if $s$ is a nonpositive integer, where $\Gamma$ has a pole.

 For $s$ a \typ{PADIC}, evaluates the Morita gamma function at $s$, that
 is the unique continuous $p$-adic function on the $p$-adic integers
 extending $\Gamma_p(k)=(-1)^k \prod_{j<k}'j$, where the prime means that $p$
 does not divide $j$.
 \bprog
 ? gamma(1/4 + O(5^10))
 %1= 1 + 4*5 + 3*5^4 + 5^6 + 5^7 + 4*5^9 + O(5^10)
 ? algdep(%,4)
 %2 = x^4 + 4*x^2 + 5
 @eprog
Variant: For a \typ{PADIC} $x$, the function \fun{GEN}{Qp_gamma}{GEN x} is
 also available.