Reciprocal Gamma function

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The reciprocal gamma function is a function over the complex plane defined as

$f(z):={\begin{cases}{\frac {1}{\Gamma (z)}}&{\text{if }}z\not \in \{0\}\cup \mathbb {Z} ^{-},\\0&{\text{if }}z\in \{0\}\cup \mathbb {Z} ^{-}.\end{cases}}$

where $\Gamma (z)$ denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function.

Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.

Formulae

$f(1+z)\,f(1-z)={\frac {\sin \pi z}{\pi z}}={\rm {sinc}}_{\pi }(z),\quad z\in \mathbb {C} ,\,$

where $\scriptstyle {\rm {sinc}}_{\pi }(z)\,:=\,{\frac {\sin \pi z}{\pi z}}\,$ is the normalized sinc function.

Taylor series

The Taylor series expansion of the reciprocal gamma function about 0 is

$f(z)=\sum _{k=0}^{\infty }a_{k}\,z^{k}=z+\gamma \,z^{2}+{\frac {1}{2}}\left(\gamma ^{2}-{\frac {\pi ^{2}}{6}}\right)z^{3}+\cdots \,$

where $\gamma$ is the Euler–Mascheroni constant and $\scriptstyle {\frac {\pi ^{2}}{6}}\,=\,\zeta (2)\,$ is obtained from the Riemann zeta function.

The Taylor series expansion coefficients obey the recursion

$a_{0}=0;\,$

$a_{1}=1;\,$

$a_{2}=\gamma ;\,$

$a_{k}=k\,a_{1}a_{k}-a_{2}a_{k-1}+\sum _{j=2}^{k}(-1)^{j}\,\zeta (j)\,a_{k-j},\quad k>2;\,$

where $\zeta (s)$ is the Riemann zeta function.

Integral along the real axis

Integration of the reciprocal gamma function along the positive real axis gives the value

$\int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx=2.807770242028519365221501186557772932308085920930198291220054809597100\ldots ,\,$

which is known as the Fransén–Robinson constant.

A058655 Decimal expansion of area under the curve 1/Gamma(x) from zero to infinity.

{2, 8, 0, 7, 7, 7, 0, 2, 4, 2, 0, 2, 8, 5, 1, 9, 3, 6, 5, 2, 2, 1, 5, 0, 1, 1, 8, 6, 5, 5, 7, 7, 7, 2, 9, 3, 2, 3, 0, 8, 0, 8, 5, 9, 2, 0, 9, 3, 0, 1, 9, 8, 2, 9, 1, 2, 2, 0, ...}