ディガンマ凾数同士の差

ディガンマ凾数同士の差

$$\begin{array}{rcl}\psi \left(y\right)-\psi \left(x\right)& =& \underset{n\to \mathrm{\infty }}{lim}\{\log \left(n\right)-\sum _{k=0}^n\frac{1}{y+k}\}-\underset{n\to \mathrm{\infty }}{lim}\{\log \left(n\right)-\sum _{k=0}^n\frac{1}{x+k}\}\dots \psi \left(x\right)=\underset{n\to \mathrm{\infty }}{lim}\{\log \left(n\right)-\sum _{k=0}^n\frac{1}{x+k}\}\\ & =& \underset{n\to \mathrm{\infty }}{lim}\{\log \left(n\right)-\sum _{k=0}^n\frac{1}{y+k}-\log \left(n\right)+\sum _{k=0}^n\frac{1}{x+k}\}\\ & =& \underset{n\to \mathrm{\infty }}{lim}\{\sum _{k=0}^n\frac{1}{x+k}-\sum _{k=0}^n\frac{1}{y+k}\}\\ & =& \sum _{k=0}^{\mathrm{\infty }}\frac{1}{x+k}-\sum _{k=0}^{\mathrm{\infty }}\frac{1}{y+k}\\ & =& {\int }_0^1\frac{u^{x-1}}{1-u}\mathrm{d}u-{\int }_0^1\frac{u^{y-1}}{1-u}\mathrm{d}u\\ & & \dots \sum _{k=0}^{\mathrm{\infty }}\frac{1}{z+k}={\int }_0^1\frac{u^{z-1}}{1-u}\mathrm{d}u\\ & =& {\int }_0^1(\frac{u^{x-1}}{1-u}-\frac{u^{y-1}}{1-u})\mathrm{d}u\\ & =& {\int }_0^1\frac{u^{x-1}-u^{y-1}}{1-u}\mathrm{d}u\end{array}$$