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*}
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逆三角関数と逆双曲線関数の冪乗積分漸化式
\[
\int\sin^{\bullet,n}xdx=x\sin^{\bullet,n}x+n\sqrt{1-x^{2}}\sin^{\bullet,n-1}x-n(n-1)\int\sin^{\bullet,n-2}xdx
\]
三角関数・双曲線関数の実部と虚部
\[
\sin z=\sin\left(\Re z\right)\cosh\left(\Im z\right)+i\cos\left(\Re z\right)\sinh\left(\Im z\right)
\]
三角関数の積
\[
\prod_{k=1}^{n-1}\sin\frac{k\pi}{n}=\frac{n}{2^{n-1}}
\]
1±itan(z)など
\[
1\pm i\tan z=\frac{1}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(e^{\pm2i\Re z}+e^{\mp2\Im z}\right)
\]