多重対数関数の漸化式
多重対数関数の漸化式
(1)
\[ Li_{s+1}'(z)=\frac{Li_{s}(z)}{z} \]
(2)
\[ Li_{s+1}(z)=\int_{0}^{z}\frac{Li_{s}(t)}{t}dt \]
(1)
\begin{align*} Li_{s+1}'(z) & =\sum_{k=1}^{\infty}\frac{kz^{k-1}}{k^{s+1}}\\ & =\frac{1}{z}\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s}}\\ & =\frac{Li_{s}(z)}{z} \end{align*}
(2)
\begin{align*} Li_{s+1}(z) & =\int_{0}^{z}Li_{s+1}'(t)dt+Li_{s+1}(0)\\ & =\int_{0}^{z}\frac{Li_{s}(t)}{t}dt \end{align*}
(2)-2
\begin{align*} Li_{s+1}(z) & =\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s+1}}\\ & =\int_{0}^{z}\sum_{k=1}^{\infty}\frac{t^{k-1}}{k^{s}}dt\\ & =\int_{0}^{z}\frac{1}{t}\sum_{k=1}^{\infty}\frac{t^{k}}{k^{s}}dt\\ & =\int_{0}^{z}\frac{Li_{s}(t)}{t}dt \end{align*}
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タイトル | 多重対数関数の漸化式 |
URL | https://www.nomuramath.com/r2j5db03/ |
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多重対数関数を含む積分
\[
\int\Li_{n}\left(z\right)dz=\sum_{k=0}^{n-2}\left\{ \left(-1\right)^{n-k}z\Li_{k+2}\left(z\right)\right\} -\left(-1\right)^{n}\left(z-\left(1-z\right)\Li_{1}\left(z\right)\right)+C
\]
多重対数関数の定義
\[
Li_{s}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s}}
\]
多重対数関数の基本的性質
\[
\Li_{1}(z)=-\log(1-z)
\]