逆正接関数・逆双曲線正接関数と多重対数関数の関係

by nomura · 2021年4月6日

逆正接関数・逆双曲線正接関数と多重対数関数の関係

(1)

\[ \Tan^{\bullet}z=\frac{i}{2}\left(-\Li_{1}\left(iz\right)+\Li_{1}\left(-iz\right)\right) \]

(2)

\[ \Tanh^{\bullet}z=\frac{1}{2}\left(\Li_{1}\left(z\right)-\Li_{1}\left(-z\right)\right) \]

(3)

\begin{align*} \int\frac{\Tan^{\bullet}z}{z}dz & =\frac{i}{2}\left(\Li_{2}\left(-iz\right)-\Li_{2}\left(iz\right)\right)+\C{} \end{align*}

(4)

\begin{align*} \int\frac{\Tanh^{\bullet}z}{z}dz & =\frac{1}{2}\left(\Li_{2}\left(z\right)-\Li_{2}\left(-z\right)\right)+\C{} \end{align*}

(1)

\begin{align*} \Tan^{\bullet}z & =\int_{0}^{z}\frac{1}{1+z^{2}}dz\\ & =\frac{1}{2}\int_{0}^{z}\left(\frac{1}{1-iz}+\frac{1}{1+iz}\right)dz\\ & =\frac{1}{2}\left(-i\Li_{1}\left(iz\right)+i\Li_{1}\left(-iz\right)\right)\\ & =\frac{i}{2}\left(-\Li_{1}\left(iz\right)+\Li_{1}\left(-iz\right)\right) \end{align*}

(2)

\begin{align*} \Tanh^{\bullet}z & =-i\Tan^{\bullet}iz\\ & =-i\left(\frac{i}{2}\left(-\Li_{1}\left(-z\right)+\Li_{1}\left(z\right)\right)\right)\\ & =\frac{1}{2}\left(\Li_{1}\left(z\right)-\Li_{1}\left(-z\right)\right) \end{align*}

(3)

\begin{align*} \int\frac{\Tan^{\bullet}z}{z}dz & =\frac{i}{2}\int\frac{\Li_{1}\left(-iz\right)-\Li_{1}\left(iz\right)}{z}dz\\ & =\frac{i}{2}\left(\Li_{2}\left(-iz\right)-\Li_{2}\left(iz\right)\right)+\C{} \end{align*}

(4)

\begin{align*} \int\frac{\Tanh^{\bullet}z}{z}dz & =-i\int\frac{\Tan^{\bullet}iz}{z}dz\\ & =-i\int^{iz}\frac{\Tan^{\bullet}z}{z}dz\\ & =-i\left[\frac{i}{2}\left(\Li_{2}\left(-iz\right)-\Li_{2}\left(iz\right)\right)\right]^{z\rightarrow iz}+\C{}\\ & =\frac{1}{2}\left(\Li_{2}\left(z\right)-\Li_{2}\left(-z\right)\right)+\C{} \end{align*}