3角関数のべき乗の積分の超幾何関数表示
3角関数のべき乗の積分の超幾何関数表示
3角関数のべき乗の積分は超幾何関数\(F\left(a,b;c,x\right)\)を使って次のように表される。
\(\alpha\in\mathbb{C}\)とする。
3角関数のべき乗の積分は超幾何関数\(F\left(a,b;c,x\right)\)を使って次のように表される。
\(\alpha\in\mathbb{C}\)とする。
(1)正弦
\begin{align*} \int\sin^{\alpha}\left(x\right)dx & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\cos\left(x\right)F\left(\frac{1-\alpha}{2},\frac{1}{2};\frac{3}{2};\cos^{2}\left(x\right)\right)+C\\ & =\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C \end{align*}(2)余弦
\begin{align*} \int\cos^{\alpha}\left(x\right)dx & =\frac{\cos^{\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\sin\left(x\right)F\left(\frac{1-\alpha}{2},\frac{1}{2};\frac{3}{2};\sin^{2}\left(x\right)\right)+C\\ & =-\frac{\left(\sin^{2}\left(x\right)\right)^{\frac{1}{2}}}{\sin\left(x\right)}\cdot\frac{\cos^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\cos^{2}\left(x\right)\right)+C \end{align*}(3)正接
\begin{align*} \int\tan^{\alpha}\left(x\right)dx & =\frac{\tan^{\alpha+1}\left(x\right)}{\alpha+1}F\left(1,\frac{\alpha+1}{2};\frac{\alpha+3}{2};-\tan^{2}\left(x\right)\right)+C\\ & =\frac{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+1}{2}}}{\cos^{\alpha+1}\left(x\right)}\cdot\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{\alpha+1}{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}}}\cdot\frac{\cos^{1-\alpha}\left(x\right)}{1-\alpha}F\left(\frac{1-\alpha}{2},\frac{1-\alpha}{2};\frac{3-\alpha}{2};\cos^{2}\left(x\right)\right)+C \end{align*}(1)
\begin{align*} \int\sin^{\alpha}\left(x\right)dx & =\int\sin^{\alpha-1}\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\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}\sin\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}}d\cos\left(x\right)\\ & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int 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}}}\cos\left(x\right)F\left(\frac{1-\alpha}{2},\frac{1}{2};\frac{3}{2};\cos^{2}\left(x\right)\right)+C \end{align*} \begin{align*} \int\sin^{\alpha}\left(x\right)dx & =\int\sin^{\alpha}\left(x\right)\cos^{-1}\left(x\right)\cos\left(x\right)dx\\ & =\frac{\cos^{-1}\left(x\right)}{\left(\cos^{-2}\left(x\right)\right)^{\frac{1}{2}}}\int\sin^{\alpha}\left(x\right)\left(\cos^{-2}\left(x\right)\right)^{\frac{1}{2}}\cos\left(x\right)dx\\ & =\frac{\left(\cos^{2}\left(x\right)\right)^{\frac{1}{2}}}{\cos\left(x\right)}\int\sin^{\alpha}\left(x\right)\left(1-\sin^{2}\left(x\right)\right)^{-\frac{1}{2}}d\sin\left(x\right)\\ & =\int\sin^{\alpha}\left(x\right)F\left(\frac{1}{2};;\sin^{2}\left(x\right)\right)d\sin\left(x\right)\\ & =\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C \end{align*}(2)
\begin{align*} \int\cos^{\alpha}\left(x\right)dx & =\int\cos^{\alpha-1}\left(x\right)\cos\left(x\right)dx\\ & =\frac{\cos^{\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}\cos\left(x\right)dx\\ & =\frac{\cos^{\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int\left(1-\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}d\sin\left(x\right)\\ & =\frac{\cos^{\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int F\left(\frac{1-\alpha}{2};;\sin^{2}\left(x\right)\right)d\sin\left(x\right)\\ & =\frac{\cos^{\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int F\left(\frac{1-\alpha}{2};;\sin^{2}\left(x\right)\right)d\sin\left(x\right)\\ & =\frac{\cos^{\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\sin\left(x\right)F\left(\frac{1-\alpha}{2},\frac{1}{2};\frac{3}{2};\sin^{2}\left(x\right)\right)+C \end{align*} \begin{align*} \int\cos^{\alpha}\left(x\right)dx & =\int\cos^{\alpha}\left(x\right)\sin^{-1}\left(x\right)\sin\left(x\right)dx\\ & =\frac{\sin^{-1}\left(x\right)}{\left(\sin^{-2}\left(x\right)\right)^{\frac{1}{2}}}\int\cos^{\alpha}\left(x\right)\left(\sin^{-2}\left(x\right)\right)^{\frac{1}{2}}\sin\left(x\right)dx\\ & =-\frac{\left(\sin^{2}\left(x\right)\right)^{\frac{1}{2}}}{\sin\left(x\right)}\int\cos^{\alpha}\left(x\right)\left(1-\cos^{2}\left(x\right)\right)^{-\frac{1}{2}}d\cos\left(x\right)\\ & =-\frac{\left(\sin^{2}\left(x\right)\right)^{\frac{1}{2}}}{\sin\left(x\right)}\int\cos^{\alpha}\left(x\right)F\left(\frac{1}{2};;\cos^{2}\left(x\right)\right)d\cos\left(x\right)\\ & =-\frac{\left(\sin^{2}\left(x\right)\right)^{\frac{1}{2}}}{\sin\left(x\right)}\cdot\frac{\cos^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\cos^{2}\left(x\right)\right)+C \end{align*}(3)
\begin{align*} \int\tan^{\alpha}\left(x\right)dx & =\int\tan^{\alpha}\left(x\right)\cos^{2}\left(x\right)\cos^{-2}\left(x\right)dx\\ & =\int\tan^{\alpha}\left(x\right)\left(1+\tan^{2}\left(x\right)\right)^{-1}d\tan\left(x\right)\\ & =\int\tan^{\alpha}\left(x\right)F\left(1;;-\tan^{2}\left(x\right)\right)d\tan\left(x\right)\\ & =\frac{\tan^{\alpha+1}\left(x\right)}{\alpha+1}F\left(1,\frac{\alpha+1}{2};\frac{\alpha+3}{2};-\tan^{2}\left(x\right)\right)+C \end{align*} \begin{align*} \int\tan^{\alpha}\left(x\right)dx & =\int\sin^{\alpha}\left(x\right)\cos^{-\alpha}\left(x\right)dx\\ & =\frac{\cos^{-\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{-\frac{\alpha+1}{2}}}\int\sin^{\alpha}\left(x\right)\left(\cos^{2}\left(x\right)\right)^{-\frac{\alpha+1}{2}}\cos\left(x\right)dx\\ & =\frac{\cos^{-\alpha-1}\left(x\right)}{\left(\cos^{2}\left(x\right)\right)^{-\frac{\alpha+1}{2}}}\int\sin^{\alpha}\left(x\right)\left(1-\sin^{2}\left(x\right)\right)^{-\frac{\alpha+1}{2}}d\sin\left(x\right)\\ & =\frac{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+1}{2}}}{\cos^{\alpha+1}\left(x\right)}\int\sin^{\alpha}\left(x\right)F\left(\frac{\alpha+1}{2};;\sin^{2}\left(x\right)\right)d\sin\left(x\right)\\ & =\frac{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+1}{2}}}{\cos^{\alpha+1}\left(x\right)}\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{\alpha+1}{2},\frac{\alpha+1}{2};\frac{\alpha+1}{2}+1;\sin^{2}\left(x\right)\right)+C\\ & =\frac{\left(\cos^{2}\left(x\right)\right)^{\frac{\alpha+1}{2}}}{\cos^{\alpha+1}\left(x\right)}\cdot\frac{\sin^{\alpha+1}\left(x\right)}{\alpha+1}F\left(\frac{\alpha+1}{2},\frac{\alpha+1}{2};\frac{\alpha+3}{2};\sin^{2}\left(x\right)\right)+C \end{align*} \begin{align*} \int\tan^{\alpha}\left(x\right)dx & =\int\sin^{\alpha}\left(x\right)\cos^{-\alpha}\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}}\cos^{-\alpha}\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\left(1-\cos^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}\cos^{-\alpha}\left(x\right)d\cos\left(x\right)\\ & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\int F\left(\frac{1-\alpha}{2};;\cos^{2}\left(x\right)\right)\cos^{-\alpha}\left(x\right)d\cos\left(x\right)\\ & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\cdot\frac{\cos^{1-\alpha}\left(x\right)}{1-\alpha}F\left(\frac{1-\alpha}{2},\frac{1-\alpha}{2};\frac{1-\alpha}{2}+1;\cos^{2}\left(x\right)\right)+C\\ & =-\frac{\sin^{\alpha-1}\left(x\right)}{\left(\sin^{2}\left(x\right)\right)^{\frac{\alpha-1}{2}}}\cdot\frac{\cos^{1-\alpha}\left(x\right)}{1-\alpha}F\left(\frac{1-\alpha}{2},\frac{1-\alpha}{2};\frac{3-\alpha}{2};\cos^{2}\left(x\right)\right)+C \end{align*}
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