ディガンマ凾数の極限表示

ディガンマ凾数

$$\begin{array}{rcl}\psi \left(z\right)& =& \frac{\mathrm{d}}{\mathrm{d}z}\log \left(\mathrm{\Gamma }\left(z\right)\right)\\ & =& \frac{{\mathrm{\Gamma }}^{\prime }\left(z\right)}{\mathrm{\Gamma }\left(z\right)}\end{array}$$

ディガンマ凾数の極限表示

$$\begin{array}{rcl}\mathrm{\Gamma }\left(z\right)& =& \underset{n\to \mathrm{\infty }}{lim}\frac{n^{z}n!}{\prod _{k=0}^n(z+k)}\\ \log \left(\mathrm{\Gamma }\left(z\right)\right)& =& \log \left(\underset{n\to \mathrm{\infty }}{lim}\frac{n^{z}n!}{\prod _{k=0}^n(z+k)}\right)\\ & =& \underset{n\to \mathrm{\infty }}{lim}\log \left(\frac{n^{z}n!}{\prod _{k=0}^n(z+k)}\right)\\ & =& \underset{n\to \mathrm{\infty }}{lim}\{\log \left(n^z\right)+\log (n!)-\log \left(\prod _{k=0}^n(z+k)\right)\}\\ & =& \underset{n\to \mathrm{\infty }}{lim}\{\log \left(n^z\right)+\log (n!)-\log (z+0)-\log (z+1)-\log (z+2)-\cdots -\log (z+n)\}\\ \psi \left(z\right)=\frac{\mathrm{d}}{\mathrm{d}z}\log \left(\mathrm{\Gamma }\left(z\right)\right)& =& \frac{\mathrm{d}}{\mathrm{d}z}\left[\underset{n\to \mathrm{\infty }}{lim}\{\log \left(n^z\right)+\log (n!)-\log (z+0)-\log (z+1)-\log (z+2)-\cdots -\log (z+n)\}\right]\\ & =& \underset{n\to \mathrm{\infty }}{lim}\left[\frac{\mathrm{d}}{\mathrm{d}z}\{\log \left(n^z\right)+\log (n!)-\log (z+0)-\log (z+1)-\log (z+2)-\cdots -\log (z+n)\}\right]\\ & =& \underset{n\to \mathrm{\infty }}{lim}\{\log \left(n\right)-\frac{1}{z}-\frac{1}{z+1}-\frac{1}{z+2}-\cdots -\frac{1}{z+n}\}\\ & =& \underset{n\to \mathrm{\infty }}{lim}[\log \left(n\right)-\{\frac{1}{z}+\frac{1}{z+1}+\frac{1}{z+2}+\cdots +\frac{1}{z+n}\}]\\ & =& \underset{n\to \mathrm{\infty }}{lim}\{\log \left(n\right)-\sum _{k=0}^n\frac{1}{z+k}\}\end{array}$$