# 倒數伽瑪函數

Γ函數的倒數
Γ函數（藍色）、Γ函數的倒數（橘色）。
Γ函數的倒數的函數圖形

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

## 無窮乘積展開

{\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}}}

## 泰勒級數

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

${\displaystyle a_{n}={\frac {a_{2}a_{n-1}-\sum _{j=2}^{n-1}(-1)^{j}\,\zeta (j)\,a_{n-j}}{n-1}}}$

${\displaystyle a_{n}={\frac {(-1)^{n}}{\pi n!}}\int _{0}^{\infty }e^{-t}\Im [(\log(t)-i\pi )^{n}]dt.}$

an的近似值為[1]

${\displaystyle a_{n}\approx (-1)^{n}{\sqrt {\frac {2}{\pi }}}{\frac {\sqrt {n}}{n!}}\Im \left(e^{-nz_{0}}{\frac {z_{0}^{1/2-n}}{\sqrt {1+z_{0}}}}\right),}$

${\displaystyle W_{-1}}$ 是分支為負一的朗伯W函数

## 漸近展開

|z|arg(z)為一固定值的情形下趨於無窮，則有：

${\displaystyle \ln(1/\Gamma (z))\sim -z\ln(z)+z+{\tfrac {1}{2}}\ln \left({\frac {z}{2\pi }}\right)-{\frac {1}{12z}}+{\frac {1}{360z^{3}}}-{\frac {1}{1260z^{5}}}\qquad \qquad {\text{for}}\quad |\arg(z)|<\pi }$

## 以圍線積分表示

${\displaystyle {\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\oint _{H}(-t)^{-z}e^{-t}\,dt,}$

## 階乘倒數

${\displaystyle \prod _{k=1}^{n}{\frac {1}{k}}={\frac {1}{n!}}\quad \forall n\geq 1}$

${\displaystyle \sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {1}{k}}=e}$

${\displaystyle {\frac {1}{n!}}={\frac {1}{\Gamma (z+1)}}}$ .

${\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{-n\imath \vartheta }e^{e^{\imath \vartheta }}\ d\vartheta }$ .

${\displaystyle {\frac {1}{\Gamma (z)}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }(c+\imath t)^{-z}e^{c+\imath t}dt,}$

${\displaystyle {\frac {1}{(2n-1)!!}}={\frac {\sqrt {\pi }}{2^{n}\cdot \Gamma \left(n+{\frac {1}{2}}\right)}}}$

## 積分

${\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx\approx 2.80777024,}$ OEIS中的数列A058655

## 參考文獻

1. Fekih-Ahmed, L. (2014). On the Power Series Expansion of the Reciprocal Gamma Function pdf (PDF).. HAL archives,
2. ^ Hazewinkel, Michiel (编), Weierstrass theorem, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 2001 [1994], ISBN 978-1-55608-010-4
3. ^ Wrench, J.W. (1968). Concerning two series for the gamma function. Mathematics of Computation, 22, 617–626. and
Wrench, J.W. (1973). Erratum: Concerning two series for the gamma function. Mathematics of Computation, 27, 681–682.
4. ^ 圍線積分 contour integration
5. ^ Iwanami Sūgaku Jiten Fourth, Tokyo: Iwanami Shoten, 2007, ISBN 978-4-00-080309-0, MR 2383190 （日语） 142.D
6. ^ Graham, Knuth, and Patashnik. Concrete Mathematics. Addison-Wesley. 1994: 566.
7. ^ Integral formula for ${\displaystyle 1/\Gamma (z)}$ . Math Stack Exchange.
8. ^ Finch, S. R. "Fransén-Robinson Constant." §4.6 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 262-264, 2003.
1. Thomas Schmelzer & Lloyd N. Trefethen, Computing the Gamma function using contour integrals and rational approximations
2. Mette Lund, An integral for the reciprocal Gamma function
3. Milton Abramowitz & Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
4. Eric W. Weisstein, Gamma Function, MathWorld