カテゴリー: 三角関数

\[ \tan^{\pm1}z=i^{\pm1}\left(1+2\Li_{0}\left(\mp e^{2iz}\right)\right) \]

\[ 1+\sin z=\left(\cos\frac{z}{2}+\sin\frac{z}{2}\right)^{2} \]

\[ \tan\frac{z}{2}=\frac{\sin z}{1+\cos z} \]

\[ \sin z=\frac{2\tan\frac{z}{2}}{1+\tan^{2}\frac{z}{2}} \]

\[ \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz=-\frac{2}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\bullet}\frac{\left(\gamma-\beta\right)\tan\frac{z}{2}+\alpha}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}+C \]

\[ \sin^{\bullet}\sin z=?z \]

\[ \sin\Arg z=\frac{\Im z}{\left|z\right|} \]

\[ \cos z\pm i\sin z=e^{\pm iz} \]

\[ 1\pm i\tan z=\frac{1}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(e^{\pm2i\Re z}+e^{\mp2\Im z}\right) \]

\[ \sin z=\sin\left(\Re z\right)\cosh\left(\Im z\right)+i\cos\left(\Re z\right)\sinh\left(\Im z\right) \]

\[ \Sin^{\bullet}\left(iz\right)=i\Sinh^{\bullet}z \]

\[ \Sin^{\bullet}\left(-z\right)=-\Sin^{\bullet}z \]

\[ \tan z=\frac{\sin\left(2\Re z\right)+i\sinh\left(2\Im z\right)}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \]

\[ \int z^{\alpha}\Sin^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sin^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C \]

\[ \Tan^{\bullet}z=\frac{i}{2}\left(-\Li_{1}\left(iz\right)+\Li_{1}\left(-iz\right)\right) \]

\[ \sin z=\sin\left(\Re\left(z\right)\right)\cosh\left(\Im\left(z\right)\right)+i\cos\left(\Re\left(z\right)\right)\sinh\left(\Im\left(z\right)\right) \]

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

\[ \left(\sin x\right)'=\cos x \]

\[ \int f(\cos x,\sin x)dx=\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt\cnd{t=\tan\frac{x}{2}} \]

\[ \Sin^{\bullet}z=-i\Log\left(iz+\sqrt{1-z^{2}}\right) \]

\[ \log\left(\sin\left(\pi x\right)\right)=\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k} \]

\[ \int\Log\sin^{\alpha}zdz=z\Log\sin^{\alpha}x+\frac{i\alpha}{2}z^{2}+\alpha z\Li_{1}\left(e^{2iz}\right)+\frac{i\alpha}{2}\Li_{2}\left(e^{2iz}\right)+\C{} \]

\[ \log\sin x=-\log2+\frac{\pi}{2}i-ix-Li_{1}\left(e^{2ix}\right) \]

\[ \int z\tan^{\pm1}\left(z\right)dz=i^{\pm1}\left\{ \frac{1}{2}z^{2}-iz\Li_{1}\left(\mp e^{2iz}\right)+\frac{1}{2}\Li_{2}\left(\mp e^{2iz}\right)\right\} +C \]

\[ \prod_{k=1}^{n-1}\sin\frac{k\pi}{n}=\frac{n}{2^{n-1}} \]

\[ \int\sin^{\bullet,n}xdx=x\sin^{\bullet,n}x+n\sqrt{1-x^{2}}\sin^{\bullet,n-1}x-n(n-1)\int\sin^{\bullet,n-2}xdx \]

\[ \int\sin^{n}xdx=-\frac{1}{n}\cos x\sin^{n-1}x+\frac{n-1}{n}\int\sin^{n-2}xdx\qquad(n\ne0) \]

\[ \int\sin^{\bullet}xdx=x\sin^{\bullet}x+\sqrt{1-x^{2}} \]

\[ \sin^{\bullet}x=\sum_{k=0}^{\infty}\frac{C\left(2k,k\right)}{4^{k}(2k+1)}x^{2k+1}\qquad,(|x|\leq1) \]

\[ \sin^{\bullet}x=\int_{0}^{x}\frac{1}{\sqrt{1-t^{2}}}dt \]