12.8 Gamma & Reciprocal Gamma Functions $\Gamma(s)$ and $\frac{1}{\Gamma(s)}$

Navigation Links: – First Page – Parent Page


The following plot illustrates a derived formula for the reciprocal Gamma function $\frac{1}{\Gamma(s)}$ converges to the reference function (dashed-gray curve) for $s\ge 0$ as the evaluation limit is successively increased (green, orange, and blue curves) .

$\frac{1}{\Gamma[s]}$

The following plot illustrates a derived formula for the Gamma function $\Gamma(s)$ (orange curve) converges to the reference function (blue curve) for $-1<s<1$.

$\Gamma(s)$

The following plot illustrates a derived formula for $\Gamma(1-s)$ (orange curve) converges to the reference function (blue curve) for $0<s<2$. The formula for $\Gamma(1-s)$ illustrated below is essentially the formula illustrated for $\Gamma(s)$ illustrated above evaluated at $1-s$. The function $\Gamma(1-s)$ is of particular interest because of it’s appearance in the Riemann zeta functional equation defined below.

(1) $\quad\zeta (s)=2^s\pi ^{s-1}\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma (1-s)\,\zeta (1-s)$

$\Gamma(1-s)$

The following two plots illustrate the real and imaginary parts of the derived formula for $\Gamma(1-s)$ (orange curves) converge to the real and imaginary parts of the reference function (blue curves) along the critical line $s=\frac{1}{2}+i\,t$ for $-3\le t\le 3$. The convergence of the real part of the derived formula for $\Gamma(1-s)$ (orange curve in the first plot below) is a bit slower for small imaginary values of $s$, but convergence increases as the formula evaluation limits increase towards infinity.

$\Re(\Gamma(1-s))$ evaluated at $s=\frac{1}{2}+i\,t$.
$\Im(\Gamma(1-s))$ evaluated at $s=\frac{1}{2}+i\,t$.