多重対数関数の基本的性質
多重対数関数の基本的性質
(1)
\[ \Li_{s}(0)=0 \](2)
\[ \Li_{0}\left(z\right)=\frac{z}{1-z} \](3)
\[ \Li_{1}(z)=-\log(1-z) \](4)多重対数関数とリーマン・ゼータ関数
\[ \Li_{s}(1)=\zeta(s) \](5)多重対数関数とディリクレ・イータ関数
\[ \Li_{s}(-1)=-\eta(s) \](1)
\begin{align*} \Li_{s}(0) & =\sum_{k=1}^{\infty}\frac{0^{k}}{k^{s}}\\ & =0 \end{align*}(2)
\begin{align*} \Li_{0}(z) & =\sum_{k=1}^{\infty}\frac{z^{k}}{k^{0}}\\ & =z\sum_{k=0}^{\infty}z^{k}\\ & =\frac{z}{1-z} \end{align*}(3)
\begin{align*} \Li_{1}(z) & =\sum_{k=1}^{\infty}\frac{\left(z\right)^{k}}{k}\\ & =-\log(1-z) \end{align*}(4)
\begin{align*} \Li_{s}(1) & =\sum_{k=1}^{\infty}\frac{1^{k}}{k^{s}}\\ & =\zeta(s) \end{align*}(5)
\begin{align*} \Li_{s}(-1) & =\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k^{s}}\\ & =-\eta(s) \end{align*}
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| タイトル | 多重対数関数の基本的性質 |
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逆数の多重対数関数
\[
\Li_{n}\left(\frac{1}{z}\right)=\left(-1\right)^{n+1}\Li_{n}\left(z\right)+\left(1+\left(-1\right)^{n}\right)\zeta\left(n\right)+\left(-1\right)^{n}\sum_{k=0}^{\left\lfloor \frac{n-3}{2}\right\rfloor }\left\{ 2\zeta\left(2\left(k+1\right)\right)\frac{\Log^{n-2\left(k+1\right)}z}{\left(n-2\left(k+1\right)\right)!}\right\} +\left(-1\right)^{n+1}\frac{\Log^{n}z}{n!}+\left(-1\right)^{n+1}\frac{\Log^{n-1}z}{\left(n-1\right)!}\left(\Log\left(1-z\right)-\Log\left(z-1\right)\right)
\]
多重対数関数の関係
\[
\Li_{n}\left(z\right)+\Li_{n}\left(-z\right)=\frac{1}{2^{n-1}}\Li_{n}\left(z^{2}\right)
\]
多重対数関数を含む積分
\[
\int\Li_{n}\left(z\right)dz=\sum_{k=0}^{n-2}\left\{ \left(-1\right)^{n-k}z\Li_{k+2}\left(z\right)\right\} -\left(-1\right)^{n}\left(z-\left(1-z\right)\Li_{1}\left(z\right)\right)+C
\]
指数関数の多重対数関数の積分
\[
\int\Li_{n}\left(e^{z}\right)dz=\Li_{n+1}\left(e^{z}\right)+C
\]