ディガンマ凾数
$$\begin{eqnarray}
\psi\left(z\right)&=&\frac{\mathrm{d}}{\mathrm{d}z} \log{\left(\Gamma\left(z\right)\right)}
\\&=&\frac{\Gamma^\prime\left(z\right)}{\Gamma\left(z\right)}
\end{eqnarray}$$
ディガンマ凾数の極限表示
$$\begin{eqnarray}
\Gamma\left(z\right)
&=&\href{https://shikitenkai.blogspot.com/2021/07/blog-post_16.html}{\lim_{n\rightarrow\infty}\frac{n^z n!}{\prod_{k=0}^n\left(z+k\right)}}
\\\log{\left(\Gamma\left(z\right)\right)}
&=&\log{\left(\lim_{n\rightarrow\infty}\frac{n^z n!}{\prod_{k=0}^n\left(z+k\right)}\right)}
\\&=&\lim_{n\rightarrow\infty}\log{\left(\frac{n^z n!}{\prod_{k=0}^n\left(z+k\right)}\right)}
\\&=&\lim_{n\rightarrow\infty} \left\{
\log{\left(n^z \right)}
+\log{\left(n! \right)}
-\log{\left(
\prod_{k=0}^n\left(z+k\right)
\right)}
\right\}
\\&=&\lim_{n\rightarrow\infty} \left\{
\log{\left(n^z \right)}
+\log{\left(n! \right)}
-\log{\left(z+0\right)}-\log{\left(z+1\right)}-\log{\left(z+2\right)}-\cdots-\log{\left(z+n\right)}
\right\}
\\\psi\left(z\right)=\frac{\mathrm{d}}{\mathrm{d}z} \log{\left(\Gamma\left(z\right)\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}z}\left[
\lim_{n\rightarrow\infty} \left\{
\log{\left(n^z \right)}
+\log{\left(n! \right)}
-\log{\left(z+0\right)}-\log{\left(z+1\right)}-\log{\left(z+2\right)}-\cdots-\log{\left(z+n\right)}
\right\}\right]
\\&=&\lim_{n\rightarrow\infty} \left[
\frac{\mathrm{d}}{\mathrm{d}z}\left\{
\log{\left(n^z \right)}
+\log{\left(n! \right)}
-\log{\left(z+0\right)}-\log{\left(z+1\right)}-\log{\left(z+2\right)}-\cdots-\log{\left(z+n\right)}
\right\}\right]
\\&=&\lim_{n\rightarrow\infty} \left\{
\log{\left(n \right)}
-\frac{1}{z}-\frac{1}{z+1}-\frac{1}{z+2}-\cdots-\frac{1}{z+n}
\right\}
\\&=&\lim_{n\rightarrow\infty} \left[
\log{\left(n \right)}
-\left\{\frac{1}{z}+\frac{1}{z+1}+\frac{1}{z+2}+\cdots+\frac{1}{z+n}\right\}
\right]
\\&=&\lim_{n\rightarrow\infty} \left\{
\log{\left(n \right)}
-\sum_{k=0}^n \frac{1}{z+k}
\right\}
\end{eqnarray}$$
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