ディガンマ凾数の相反公式

ディガンマ凾数の相反公式

ディガンマ凾数の定義

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

\(1-z\)のディガンマ凾数

$$\begin{eqnarray} \psi\left(1-z\right) &=&\frac{\mathrm{d}}{\mathrm{d}z} \log{\left(\Gamma\left(1-z\right)\right)} \\&=&\frac{\mathrm{d}}{\mathrm{d}u} \log{\left(\Gamma\left(u\right)\right)}\frac{\mathrm{d}u}{\mathrm{d}z} \;\ldots\;u=1-z,\;\frac{\mathrm{d}u}{\mathrm{d}z}=-1 \\&=&-\frac{\mathrm{d}}{\mathrm{d}z}\log\left(\Gamma\left(1-z\right)\right) \\&=&-\frac{\mathrm{d}}{\mathrm{d}z}\log\left(\frac{\Gamma\left(z\right)}{\Gamma\left(z\right)}\Gamma\left(1-z\right)\right) \\&=&-\frac{\mathrm{d}}{\mathrm{d}z}\log\left(\frac{1}{\Gamma\left(z\right)}\frac{\pi}{\sin{\left(\pi z\right)}}\right) \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/blog-post_26.html}{\Gamma\left(z\right)\Gamma\left(1-z\right)=\frac{\pi}{\sin{\left(\pi z\right)}}} \\&=&-\left\{ \frac{\mathrm{d}}{\mathrm{d}z} \log \left(\pi\right) -\frac{\mathrm{d}}{\mathrm{d}z} \log \left(\sin{\left(\pi z\right)}\right) -\frac{\mathrm{d}}{\mathrm{d}z} \log \left(\Gamma\left(z\right)\right) \right\} \\&=&-\left\{0-\pi\cot{\left(\pi z\right)}-\psi\left(z\right)\right\} \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/logsinz.html}{\frac{\mathrm{d}}{\mathrm{d}z} \log \sin{\left(\pi z\right)}=\pi\cot{\left(\pi z\right)}} \\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}z} \log \Gamma\left(z\right)=\psi\left(z\right),\;(ディガンマ凾数の定義) \\&=&\pi\cot{\left(\pi z\right)}+\psi\left(z\right) \end{eqnarray}$$

ディガンマ凾数の相反公式

$$\begin{eqnarray} \psi\left(1-z\right) &=&\pi\cot{\left(\pi z\right)}+\psi\left(z\right) \\\psi\left(1-z\right)-\psi\left(z\right)&=&\pi\cot{\left(\pi z\right)} \end{eqnarray}$$