巾関数と逆三角関数・逆双曲線関数の積の積分
巾関数と逆三角関数の積の積分
(1)
\[ \int z^{\alpha}\Sin^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sin^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \](2)
\[ \int z^{\alpha}\Cos^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Cos^{\bullet}z+\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \](3)
\[ \int z^{\alpha}\Tan^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Tan^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;-z^{2}\right)\right)+C \]-
\(F\)は一般化超幾何関数(1)
\begin{align*} \int z^{\alpha}\Sin^{\bullet}zdz & =\frac{1}{\alpha+1}z^{\alpha+1}\Sin^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}\frac{1}{\sqrt{1-z^{2}}}dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Sin^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}F\left(\frac{1}{2};;z^{2}\right)dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Sin^{\bullet}z-\frac{1}{\alpha+1}\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha+2}{2};\frac{\alpha+2}{2}+1;z^{2}\right)+C\\ & =\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 \end{align*}(2)
\begin{align*} \int z^{\alpha}\Cos^{\bullet}zdz & =\frac{1}{\alpha+1}z^{\alpha+1}\Cos^{\bullet}z+\frac{1}{\alpha+1}\int z^{\alpha+1}\frac{1}{\sqrt{1-z^{2}}}dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Cos^{\bullet}z+\frac{1}{\alpha+1}\int z^{\alpha+1}F\left(\frac{1}{2};;z^{2}\right)dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Cos^{\bullet}z+\frac{1}{\alpha+1}\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha+2}{2};\frac{\alpha+2}{2}+1;z^{2}\right)+C\\ & =\frac{1}{\alpha+1}\left(z^{\alpha+1}\Cos^{\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 \end{align*}(3)
\begin{align*} \int z^{\alpha}\Tan^{\bullet}zdz & =\frac{z^{\alpha+1}}{\alpha+1}\Tan^{\bullet}z-\frac{1}{\alpha+1}\int\frac{z^{\alpha+1}}{1+z^{2}}dz\\ & =\frac{z^{\alpha+1}}{\alpha+1}\Tan^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}F\left(1;;-z^{2}\right)dz\\ & =\frac{z^{\alpha+1}}{\alpha+1}\Tan^{\bullet}z-\frac{1}{\alpha+1}\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\frac{\alpha+2}{2};\frac{\alpha+2}{2}+1;-z^{2}\right)+C\\ & =\frac{1}{\alpha+1}\left(z^{\alpha+1}\Tan^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;-z^{2}\right)\right)+C \end{align*}巾関数と逆双曲線関数の積の積分
(1)
\[ \int z^{\alpha}\Sinh^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sinh^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;-z^{2}\right)\right)+C \](2)
\[ \int z^{\alpha}\Cosh^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Cosh^{\bullet}z-\frac{\sqrt{1-z^{2}}}{\sqrt{z-1}\sqrt{z+1}}\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \](3)
\[ \int z^{\alpha}\Tanh^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Tanh^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \]-
\(F\)は一般化超幾何関数(1)
\begin{align*} \int z^{\alpha}\Sinh^{\bullet}zdz & =\frac{1}{\alpha+1}z^{\alpha+1}\Sinh^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}\frac{1}{\sqrt{1+z^{2}}}dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Sinh^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}F\left(\frac{1}{2};;-z^{2}\right)dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Sinh^{\bullet}z-\frac{1}{\alpha+1}\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha+2}{2};\frac{\alpha+2}{2}+1;-z^{2}\right)+C\\ & =\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sinh^{\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 \end{align*}(2)
\begin{align*} \int z^{\alpha}\Cosh^{\bullet}zdz & =\frac{1}{\alpha+1}z^{\alpha+1}\Cosh^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}\frac{1}{\sqrt{z-1}\sqrt{z+1}}dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Cosh^{\bullet}z-\frac{1}{\alpha+1}\frac{\sqrt{1-z^{2}}}{\sqrt{z-1}\sqrt{z+1}}\int z^{\alpha+1}\frac{1}{\sqrt{1-z^{2}}}dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Cosh^{\bullet}z-\frac{1}{\alpha+1}\frac{\sqrt{1-z^{2}}}{\sqrt{z-1}\sqrt{z+1}}\int z^{\alpha+1}F\left(\frac{1}{2};;z^{2}\right)dz\\ & =\frac{1}{\alpha+1}z^{\alpha+1}\Cosh^{\bullet}z-\frac{1}{\alpha+1}\frac{\sqrt{1-z^{2}}}{\sqrt{z-1}\sqrt{z+1}}\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha+2}{2};\frac{\alpha+2}{2}+1;z^{2}\right)+C\\ & =\frac{1}{\alpha+1}\left(z^{\alpha+1}\Cosh^{\bullet}z-\frac{\sqrt{1-z^{2}}}{\sqrt{z-1}\sqrt{z+1}}\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \end{align*}(3)
\begin{align*} \int z^{\alpha}\Tanh^{\bullet}zdz & =\frac{z^{\alpha+1}}{\alpha+1}\Tanh^{\bullet}z-\frac{1}{\alpha+1}\int\frac{z^{\alpha+1}}{1-z^{2}}dz\\ & =\frac{z^{\alpha+1}}{\alpha+1}\Tanh^{\bullet}z-\frac{1}{\alpha+1}\int z^{\alpha+1}F\left(1;;z^{2}\right)dz\\ & =\frac{z^{\alpha+1}}{\alpha+1}\Tanh^{\bullet}z-\frac{1}{\alpha+1}\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\frac{\alpha+2}{2};\frac{\alpha+2}{2}+1;z^{2}\right)+C\\ & =\frac{1}{\alpha+1}\left(z^{\alpha+1}\Tanh^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \end{align*}ページ情報
| タイトル | 巾関数と逆三角関数・逆双曲線関数の積の積分 |
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逆正接関数・逆双曲線正接関数と多重対数関数の関係
\[
\Tan^{\bullet}z=\frac{i}{2}\left(-\Li_{1}\left(iz\right)+\Li_{1}\left(-iz\right)\right)
\]
逆三角関数と逆双曲線関数の積分
\[
\int\sin^{\bullet}xdx=x\sin^{\bullet}x+\sqrt{1-x^{2}}
\]
正接関数・双曲線正接関数の半角公式の別表示
\[
\tan\frac{z}{2}=\frac{\sin z}{1+\cos z}
\]
逆三角関数と逆双曲線関数の対数表示
\[
\Sin^{\bullet}z=-i\Log\left(iz+\sqrt{1-z^{2}}\right)
\]