逆正接関数・逆双曲線正接関数と多重対数関数の関係
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*}
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| タイトル | 逆正接関数・逆双曲線正接関数と多重対数関数の関係 |
| URL | https://www.nomuramath.com/vz64908m/ |
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三角関数と双曲線関数の冪乗積分漸化式
\[
\int\sin^{n}xdx=-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int\sin^{n-2}xdx\qquad(n\ne0)
\]
ピタゴラスの基本三角関数公式
\[
\cos^{2}x+\sin^{2}x=1
\]
偏角の3角関数
\[
\sin\Arg z=\frac{\Im z}{\left|z\right|}
\]
巾関数と逆三角関数・逆双曲線関数の積の積分
\[
\int z^{\alpha}\Sin^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sin^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C
\]