調和数・一般化調和数の積分

by nomura · 2025年5月15日

調和数・一般化調和数の積分
調和数\(H_{z}\)と一般化調和数\(H_{z,m}\)の積分は次のようになります。

(1)調和数の積分

\[ \int H_{z}dz=\log\Gamma\left(z+1\right)+\gamma z+C \]

(2)一般化調和数の積分

\begin{align*} \int H_{z,m}dz & =\zeta\left(m\right)z-\frac{\left(-1\right)^{m}}{\left(m-1\right)!}\psi^{\left(m-2\right)}\left(z+1\right)+C\\ & =\zeta\left(m\right)z+\frac{\zeta\left(m-1,z+1\right)}{m-1}+C \end{align*}

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\(\gamma\)はオイラー・マスケローニ定数
\(\psi^{\left(n\right)}\left(z\right)\)はポリガンマ関数
\(\zeta\left(s\right)\)はリーマン・ゼータ関数
\(\zeta\left(s,q\right)\)はフルヴィッツ・ゼータ関数

(1)

\begin{align*} \int H_{z}dz & =\int\left(\psi\left(z+1\right)+\gamma\right)dz\\ & =\int\left(\frac{dz}{d\left(z+1\right)}\frac{d}{dz}\log\Gamma\left(z+1\right)+\gamma\right)dz\\ & =\log\Gamma\left(z+1\right)+\gamma z+C \end{align*}

(1)-2

\begin{align*} \int H_{z}dz & =\int z\sum_{k=1}^{\infty}\frac{1}{k\left(k+z\right)}dz\\ & =\int\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{\left(k+z\right)}\right)dz\\ & =\sum_{k=1}^{\infty}\left(\frac{z}{k}-\log\left(k+z\right)+C\right)\\ & =\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\left(\frac{z}{k}-\frac{z}{n}\log n+\frac{z}{n}\log n-\log\left(k+z\right)+C\right)\\ & =\lim_{n\rightarrow\infty}\left\{ z\left(\sum_{k=1}^{n}\frac{1}{k}-\log n\right)+\log\left(n^{z}\prod_{k=1}^{n}\left(k+z\right)^{-1}\right)+C\right\} \\ & =z\gamma+\log\left(z\lim_{n\rightarrow\infty}n^{z}\prod_{k=0}^{n}\left(k+z\right)^{-1}\right)+C\\ & =z\gamma+\log\left(z\lim_{n\rightarrow\infty}\frac{\Gamma\left(z\right)}{n!}\right)+C\cmt{\Gamma\left(z\right)=\lim_{n\rightarrow\infty}n^{z}n!\prod_{k=0}^{n}\left(k+z\right)^{-1}}\\ & =z\gamma+\log\Gamma\left(z+1\right)-\log\lim_{n\rightarrow\infty}\left(n!\right)+C\\ & =z\gamma+\log\Gamma\left(z+1\right)+C \end{align*}

(2)

\begin{align*} \int H_{z,m}dz & =\int\left(\zeta\left(m\right)-\frac{\left(-1\right)^{m}}{\left(m-1\right)!}\psi^{\left(m-1\right)}\left(z+1\right)\right)dz\\ & =\zeta\left(m\right)z-\frac{\left(-1\right)^{m}}{\left(m-1\right)!}\psi^{\left(m-2\right)}\left(z+1\right)+C\tag{(*)}\\ & =\zeta\left(m\right)z-\frac{\left(-1\right)^{m}}{\left(m-1\right)!}\left(\left(-1\right)^{m-1}\left(m-2\right)!\left(\zeta\left(m-1\right)-H_{z,m-1}\right)\right)+C\cmt{\because\psi^{\left(n\right)}\left(z\right)=\left(-1\right)^{n+1}n!\left(\zeta\left(n+1\right)-H_{z-1,n+1}\right)}\\ & =\zeta\left(m\right)z+\frac{1}{m-1}\left(\zeta\left(m-1\right)-H_{z,m-1}\right)+C\\ & =\zeta\left(m\right)z+\frac{\zeta\left(m-1,z+1\right)}{m-1}+C\cmt{\because H_{z,m}=\zeta\left(m\right)-\zeta\left(m,z+1\right)} \end{align*}

(2)-2

片方のみ示す。
\begin{align*} \int H_{z,m}dz & =\int\left(\zeta\left(m\right)-\sum_{k=1}^{\infty}\frac{1}{\left(z+k\right)^{m}}\right)dz\\ & =\zeta\left(m\right)z+\frac{1}{\left(m-1\right)}\sum_{k=1}^{\infty}\frac{1}{\left(z+k\right)^{m-1}}+C\\ & =\zeta\left(m\right)z+\frac{\zeta\left(m-1,z+1\right)}{m-1}+C \end{align*}