カテゴリー: 三角関数

三角関数と双曲線関数のn乗積分

\[ \int\sin^{2n+m_{\pm}}xdx=\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}xdx\right\} \]

x tan(x)とx tanh(x)の積分

\[ \int x\tan^{\pm1}\left(x\right)=\left(i\right)^{\pm1}\left(\frac{x^{2}}{2}-ixLi_{1}\left(\mp e^{2ix}\right)+\frac{1}{2}Li_{2}\left(\mp e^{2ix}\right)\right) \]