ガンマ凾数の極限表示

ガンマ凾数の極限表示

ガンマ凾数の定義と極限表示

$$\begin{array}{rcl}\mathrm{\Gamma }\left(z\right)& =& {\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}\mathrm{d}t(\mathrm{Re}\left(z\right)>0)\dots \mathrm{定}\mathrm{義}\\ & =& \underset{n\to \mathrm{\infty }}{lim}\frac{n^{z}n!}{\prod _{k=0}^n(z+k)}\dots \mathrm{極}\mathrm{限}\mathrm{表}\mathrm{示}\end{array}$$

$G_n$の導入

$$\begin{array}{rcl}{G}_n\left(z\right)& =& {\int }_0^n{t}^{z-1}{(1-\frac{t}{n})}^n\mathrm{d}t\end{array}$$

$G_n$の総乗表示

$$\begin{array}{rcl}{G}_n\left(z\right)& =& {\int }_0^n{t}^{z-1}{(1-\frac{t}{n})}^n\mathrm{d}t\\ & =& {\int }_0^1{\left(nu\right)}^{z-1}{(1-\frac{nu}{n})}^{n}n\mathrm{d}u\\ & & \dots t=nu,u=\frac{t}{n},\frac{\mathrm{d}t}{\mathrm{d}u}=n,\mathrm{d}t=n\mathrm{d}u\\ & & \dots t:0\to n,u:0\to 1\\ & =& n^z{\int }_0^1{u}^{z-1}{(1-u)}^n\mathrm{d}u\\ & =& n^z\cdot g_n\left(z\right)\dots g_n\left(z\right)={\int }_0^1{u}^{z-1}{(1-u)}^n\mathrm{d}u\\ g_0\left(z\right)& =& {\int }_0^1{u}^{z-1}{(1-u)}^0\mathrm{d}u\\ & =& {\int }_0^1{u}^{z-1}\mathrm{d}u\\ & =& {\left[\frac{u^z}{z}\right]}_{u=0}^1\\ & =& \frac{1}{z}\\ g_n\left(z\right)& =& {\int }_0^1{u}^{z-1}{(1-u)}^n\mathrm{d}u\\ & =& {\int }_0^1{\left(\frac{u^z}{z}\right)}^{\prime }{(1-u)}^n\mathrm{d}u\dots \frac{\mathrm{d}}{\mathrm{d}u}\frac{u^z}{z}=\frac{1}{z}z{u}^{z-1}=u^{z-1}\\ & =& {\left[\frac{u^z}{z}{(1-u)}^n\right]}_{u=0}^1-{\int }_0^1\frac{u^z}{z}{\left({(1-u)}^n\right)}^{\prime }\mathrm{d}u\\ & =& [\frac{1^z}{z}{(1-1)}^n-\frac{0^z}{z}{(1-0)}^n]-{\int }_0^1\frac{u^z}{z}(n{(1-u)}^{n-1}(-1))\mathrm{d}u\\ & =& 0+\frac{n}{z}{\int }_0^1{u}^z{(1-u)}^{n-1}\mathrm{d}u\\ & =& \frac{n}{z}{g}_{n-1}(z+1)\\ & =& \frac{n}{z}\frac{n-1}{z+1}{g}_{n-2}(z+2)\\ & =& \frac{n}{z}\frac{n-1}{z+1}\frac{n-2}{z+2}{g}_{n-3}(z+3)\\ & =& \frac{n}{z}\frac{n-1}{z+1}\frac{n-2}{z+2}\dots \frac{n-(n-1)}{z+(n-1)}{g}_{n-n}(z+n)\\ & =& \frac{n}{z}\frac{n-1}{z+1}\frac{n-2}{z+2}\dots \frac{1}{z+(n-1)}{g}_0(z+n)\\ & =& \frac{n}{z}\frac{n-1}{z+1}\frac{n-2}{z+2}\dots \frac{1}{z+(n-1)}\frac{1}{z+n}\\ & =& \frac{n!}{\prod _{k=0}^n(z+k)}\\ G_n\left(z\right)& =& n^z\cdot g_n\left(z\right)\\ & =& n^z\frac{n!}{\prod _{k=0}^n(z+k)}\\ & =& \frac{n^{z}n!}{\prod _{k=0}^n(z+k)}\end{array}$$

$G_n$の極限

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}{G}_n\left(z\right)& =& \underset{n\to \mathrm{\infty }}{lim}{\int }_0^n{t}^{z-1}{(1-\frac{t}{n})}^n\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}\mathrm{d}t\\ & & \dots \underset{n\to \mathrm{\infty }}{lim}{(1-\frac{t}{n})}^n=\underset{n\to \mathrm{\infty }}{lim}{(1+\frac{-t}{n})}^n=e^{-t}\\ & =& \mathrm{\Gamma }\left(z\right)\end{array}$$

よって$G_n$の総乗表示での極限も$\mathrm{\Gamma }\left(z\right)$

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}{G}_n\left(z\right)& =& \underset{n\to \mathrm{\infty }}{lim}\frac{n^{z}n!}{\prod _{k=0}^n(z+k)}\\ & =& \mathrm{\Gamma }\left(z\right)\end{array}$$