In mathematics, the reciprocal gamma function is the function

${\displaystyle f(z)={\frac {1}{\Gamma (z)}},}$

where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log |1/Γ(z)| grows no faster than log |z|), but of infinite type (meaning that log |1/Γ(z)| grows faster than any multiple of |z|, since its growth is approximately proportional to |z| log |z| in the left-half plane).

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.

Infinite product expansion

Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infinite product expansion for the reciprocal gamma function:

${\displaystyle {\begin{aligned}{\frac {1}{\Gamma (z)}}&=z\prod _{n=1}^{\infty }{\frac {1+{\frac {z}{n}}}{\left(1+{\frac {1}{n}}\right)^{z}}}\\{\frac {1}{\Gamma (z)}}&=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)e^{-{\frac {z}{n}}}\end{aligned}}}$

where γ = 0.577216... is the Euler–Mascheroni constant. These expansions are valid for all complex numbers z.

Taylor series

Taylor series expansion around 0 gives:^[1]

${\displaystyle {\frac {1}{\ \Gamma (z)\ }}=z+\gamma \ z^{2}+\left({\frac {\gamma ^{2}}{2}}-{\frac {\pi ^{2}}{12}}\right)\ z^{3}+\left({\frac {\gamma ^{3}}{6}}-{\frac {\gamma \pi ^{2}}{12}}+{\frac {\zeta (3)}{3}}\ \right)z^{4}+\cdots \ }$

where γ is the Euler–Mascheroni constant. For n > 2, the coefficient a_n for the zⁿ term can be computed recursively as^[2][3]

${\displaystyle a_{n}={\frac {\ {a_{2}\ a_{n-1}+\sum _{j=2}^{n-1}(-1)^{j+1}\ \zeta (j)\ a_{n-j}}\ }{n-1}}={\frac {\ \gamma \ a_{n-1}-\zeta (2)\ a_{n-2}+\zeta (3)\ a_{n-3}-\cdots \ }{n-1}}}$

where ζ is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):^[3]

${\displaystyle a_{n}={\frac {(-1)^{n}}{\pi n!}}\int _{0}^{\infty }e^{-t}\ \operatorname {Im} {\Bigl }\ \mathrm {d} t~.}$

For small values, these give the following values:

Fekih-Ahmed (2014)^[3] also gives an approximation for ${\displaystyle a_{n}}$:

${\displaystyle a_n\approx <!--\; \approx \; -->\frac{(-<!--\; -\; -->1{)}^n}{(n-<!--\; -\; -->1)!}\sqrt{\frac{2}{\mathrm{\pi <!--\; \pi \; -->}n}}\mathrm{Im}<!--\; \; -->\left(\frac{z_0^{\left(1/2-<!--\; -\; -->n\right)}{e}^{-<!--\; -\; -->n{z}_0}}{\sqrt{1+z_0}}\right)}$Zdroj:https://en.wikipedia.org?pojem=Reciprocal_gamma_function
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