正弦と余弦のべき乗の積の積分の超幾何関数表示
正弦と余弦のべき乗の積の積分の超幾何関数表示
正弦と余弦のべき乗の積の積は超幾何関数\(F\left(a,b;c;x\right)\)を使って次のようになる。
\(\alpha,\beta\in\mathbb{C}\)とする。
\begin{align*} \int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx & =\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1-\beta}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C\\ & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\frac{\cos^{\beta+1}\left(x\right)}{\beta+1}F\left(\frac{1-\alpha}{2},\frac{\beta+1}{2};\frac{\beta+3}{2};\cos^{2}\left(x\right)\right)+C\\ & =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\frac{\tan^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{\alpha+\beta+2}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};-\tan^{2}\left(x\right)\right)+C\\ & =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\frac{\tan^{-\left(\beta+1\right)}}{\beta+1}\left(x\right)F\left(\frac{\alpha+\beta+2}{2},\frac{\beta+1}{2};\frac{\beta+3}{2};-\tan^{-2}\left(x\right)\right)+C \end{align*}
正弦と余弦のべき乗の積の積は超幾何関数\(F\left(a,b;c;x\right)\)を使って次のようになる。
\(\alpha,\beta\in\mathbb{C}\)とする。
\begin{align*} \int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx & =\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1-\beta}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C\\ & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\frac{\cos^{\beta+1}\left(x\right)}{\beta+1}F\left(\frac{1-\alpha}{2},\frac{\beta+1}{2};\frac{\beta+3}{2};\cos^{2}\left(x\right)\right)+C\\ & =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\frac{\tan^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{\alpha+\beta+2}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};-\tan^{2}\left(x\right)\right)+C\\ & =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\frac{\tan^{-\left(\beta+1\right)}}{\beta+1}\left(x\right)F\left(\frac{\alpha+\beta+2}{2},\frac{\beta+1}{2};\frac{\beta+3}{2};-\tan^{-2}\left(x\right)\right)+C \end{align*}
\begin{align*}
\int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx & =\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\int\sin^{\alpha}\left(x\right)\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}\cos\left(x\right)dx\\
& =\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\int\sin^{\alpha}\left(x\right)\left(1-\sin^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}\cos\left(x\right)dx\\
& =\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\int\sin^{\alpha}\left(x\right)F\left(\frac{1-\beta}{2};;\sin^{2}\left(x\right)\right)d\sin\left(x\right)\\
& =\frac{\cos^{\beta-1}}{\left(\cos^{2}\left(x\right)\right)^{\frac{\beta-1}{2}}}\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1-\beta}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C
\end{align*}
\begin{align*}
\int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx & =\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}\sin\left(x\right)\cos^{\beta}\left(x\right)dx\\
& =\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int\left(1-\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}\cos^{\beta}\left(x\right)\sin\left(x\right)dx\\
& =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int\cos^{\beta}\left(x\right)F\left(\frac{1-\alpha}{2};;\cos^{2}\left(x\right)\right)d\cos\left(x\right)\\
& =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\frac{\cos^{\beta+1}\left(x\right)}{\beta+1}F\left(\frac{1-\alpha}{2},\frac{\beta+1}{2};\frac{\beta+3}{2};\cos^{2}\left(x\right)\right)+C
\end{align*}
\begin{align*}
\int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx & =\int\sin^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)\cos^{-2}\left(x\right)dx\\
& =\int\frac{\sin^{\alpha}\left(x\right)}{\cos^{\alpha}\left(x\right)}\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)\cos^{-2}\left(x\right)dx\\
& =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\cos^{\alpha+\beta+2}\left(x\right)}\int\tan^{\alpha}\left(x\right)\cos^{\alpha+\beta+2}\left(x\right)d\tan\left(x\right)\\
& =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\cos^{\alpha+\beta+2}\left(x\right)}\frac{\cos^{\alpha+\beta+2}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\int\tan^{\alpha}\left(x\right)\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}d\tan\left(x\right)\\
& =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\int\tan^{\alpha}\left(x\right)\left(1+\tan^{2}\left(x\right)\right)^{-\frac{\alpha+\beta+2}{2}}d\tan\left(x\right)\\
& =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\int\tan^{\alpha}\left(x\right)F\left(\frac{\alpha+\beta+2}{2};;-\tan^{2}\left(x\right)\right)d\tan\left(x\right)\\
& =\frac{\cos^{\alpha}\left(x\right)\cos^{\beta+2}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\frac{\tan^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{\alpha+\beta+2}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};-\tan^{2}\left(x\right)\right)+C
\end{align*}
\begin{align*}
\int\sin^{\alpha}\left(x\right)\cos^{\beta}\left(x\right)dx & =\int\sin^{\alpha+2}\left(x\right)\cos^{\beta}\left(x\right)\sin^{-2}\left(x\right)dx\\
& =\int\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)\frac{\cos^{\beta}\left(x\right)}{\sin^{\beta}\left(x\right)}\sin^{-2}\left(x\right)dx\\
& =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\sin^{\alpha+\beta+2}\left(x\right)}\int\sin^{\alpha+\beta+2}\left(x\right)\tan^{-\beta}\left(x\right)d\tan^{-1}\left(x\right)\\
& =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\sin^{\alpha+\beta+2}\left(x\right)}\frac{\sin^{\alpha+\beta+2}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\int\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}\tan^{-\beta}\left(x\right)d\tan^{-1}\left(x\right)\\
& =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\int\left(1+\tan^{-2}\left(x\right)\right)^{-\frac{\alpha+\beta+2}{2}}\tan^{-\beta}\left(x\right)d\tan^{-1}\left(x\right)\\
& =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\int\tan^{-\beta}\left(x\right)F\left(\frac{\alpha+\beta+2}{2};;-\tan^{-2}\left(x\right)\right)d\tan^{-1}\left(x\right)\\
& =-\frac{\sin^{\alpha+2}\left(x\right)\sin^{\beta}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha+\beta+2}{2}}}\frac{\tan^{-\left(\beta+1\right)}}{\beta+1}\left(x\right)F\left(\frac{\alpha+\beta+2}{2},\frac{\beta+1}{2};\frac{\beta+3}{2};-\tan^{-2}\left(x\right)\right)+C
\end{align*}
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