第2種不完全ガンマ関数とガンマ関数の比の極限
第2種不完全ガンマ関数とガンマ関数の比の極限
(1)
\[ \lim_{k\rightarrow0}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}=\delta_{0x} \](2)
\[ \lim_{k\rightarrow\infty}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}=1 \]-
\(\Gamma\left(x\right)\)はガンマ関数、\(\Gamma\left(k,x\right)\)は第2種不完全ガンマ関数、\(\delta_{ij}\)はクロネッカーのデルタ(1)
\begin{align*} \lim_{k\rightarrow0}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)} & =\lim_{k\rightarrow0}\frac{k\Gamma\left(k,x\right)}{k\Gamma\left(k\right)}\\ & =\lim_{k\rightarrow0}\frac{k\Gamma\left(k,x\right)}{\Gamma\left(k+1\right)}\\ & =\lim_{k\rightarrow0}\frac{\Gamma\left(k+1,x\right)-x^{k}e^{-x}}{\Gamma\left(k+1\right)}\\ & =\lim_{k\rightarrow0}\frac{\Gamma\left(1,x\right)-x^{k}e^{-x}}{\Gamma\left(1\right)}\\ & =e^{-x}-\left(1-\delta_{0x}\right)e^{-x}\\ & =\delta_{0x}e^{-x}\\ & =\delta_{0x} \end{align*}(2)
\begin{align*} \lim_{k\rightarrow\infty}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)} & =\lim_{k\rightarrow0}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}+\sum_{k=+0}^{\infty}\left(\frac{\Gamma\left(k+1,x\right)}{\Gamma\left(k+1\right)}-\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}\right)\\ & =\delta_{0x}+\sum_{k=+0}^{\infty}\left(\frac{\Gamma\left(k+1,x\right)}{\Gamma\left(k+1\right)}-\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}\right)\\ & =\delta_{0x}+e^{-x}\sum_{k=+0}^{\infty}\frac{x^{k}}{k!}\\ & =\delta_{0x}+e^{-x}\left(e^{x}-\delta_{0x}\right)\\ & =\delta_{0x}\left(1-e^{-x}\right)+1\\ & =1 \end{align*}
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そのままだとΓ(0)になる積分
\[
\int_{0}^{\infty}\left(x^{-1}e^{-x}-\frac{e^{-nx}}{1-e^{-x}}\right)dx=H_{n-1}-\gamma
\]
1次式の総乗と階乗
\[
\prod_{k=a}^{b}\left(kn+r\right)=n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)}
\]
ポリガンマ(ディガンマ)関数の乗法公式
\[
\psi^{\left(m\right)}\left(nz\right)=\delta_{0,m}\log n+\frac{1}{n^{m+1}}\sum_{k=0}^{n-1}\psi^{\left(m\right)}\left(z+\frac{k}{n}\right)
\]
ガンマ関数の相反公式
\[
\Gamma(z)\Gamma(1-z)=\pi\sin^{-1}(\pi z)
\]