ディガンマ凾数同士の差

ディガンマ凾数同士の差

$$\begin{eqnarray} \psi\left(y\right)-\psi\left(x\right) &=&\lim_{n\rightarrow\infty} \left\{ \log{\left(n \right)} -\sum_{k=0}^n \frac{1}{y+k} \right\} - \lim_{n\rightarrow\infty} \left\{ \log{\left(n \right)} -\sum_{k=0}^n \frac{1}{x+k} \right\} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/blog-post_47.html}{\psi\left(x\right)=\lim_{n\rightarrow\infty} \left\{ \log{\left(n \right)} -\sum_{k=0}^n \frac{1}{x+k} \right\}} \\&=&\lim_{n\rightarrow\infty}\left\{ \log{\left(n \right)} -\sum_{k=0}^n \frac{1}{y+k} -\log{\left(n \right)} +\sum_{k=0}^n \frac{1}{x+k} \right\} \\&=&\lim_{n\rightarrow\infty}\left\{ \sum_{k=0}^n \frac{1}{x+k} -\sum_{k=0}^n \frac{1}{y+k} \right\} \\&=& \sum_{k=0}^\infty \frac{1}{x+k} -\sum_{k=0}^\infty \frac{1}{y+k} \\&=& \int_{0}^1 \frac{u^{x-1}}{1-u} \mathrm{d}u -\int_{0}^1 \frac{u^{y-1}}{1-u} \mathrm{d}u \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/uz-11-u011zk.html}{\sum_{k=0}^\infty \frac{1}{z+k}=\int_{0}^1 \frac{u^{z-1}}{1-u} \mathrm{d}u} \\&=& \int_{0}^1 \left(\frac{u^{x-1}}{1-u} - \frac{u^{y-1}}{1-u} \right)\mathrm{d}u \\&=& \int_{0}^1 \frac{u^{x-1}-u^{y-1}}{1-u}\mathrm{d}u \end{eqnarray}$$