三角関数と双曲線関数の実部と虚部
三角関数の実部と虚部
(1)
\[ \sin z=\sin\left(\Re z\right)\cosh\left(\Im z\right)+i\cos\left(\Re z\right)\sinh\left(\Im z\right) \](2)
\[ \cos z=\cos\left(\Re z\right)\cosh\left(\Im z\right)-i\sin\left(\Re z\right)\sinh\left(\Im z\right) \](3)
\[ \tan z=\frac{\sin\left(2\Re z\right)+i\sinh\left(2\Im z\right)}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \](4)
\[ \sin^{-1}z=2\frac{\sin\left(\Re z\right)\cosh\left(\Im z\right)-i\cos\left(\Re z\right)\sinh\left(\Im z\right)}{-\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \](5)
\[ \cos^{-1}z=2\frac{\cos\left(\Re z\right)\cosh\left(\Im z\right)+i\sin\left(\Re z\right)\sinh\left(\Im z\right)}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \](6)
\[ \tan^{-1}z=\frac{\sin\left(2\Re z\right)-i\sinh\left(2\Im z\right)}{-\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \](1)
\begin{align*} \sin z & =\sin\left(\Re z+i\Im z\right)\\ & =\sin\left(\Re z\right)\cos\left(i\Im z\right)+\cos\left(\Re z\right)\sin\left(i\Im z\right)\\ & =\sin\left(\Re z\right)\cosh\left(\Im z\right)+i\cos\left(\Re z\right)\sinh\left(\Im z\right) \end{align*}(2)
\begin{align*} \cos z & =\cos\left(\Re z+i\Im z\right)\\ & =\cos\left(\Re z\right)\cos\left(i\Im z\right)-\sin\left(\Re z\right)\sin\left(i\Im z\right)\\ & =\cos\left(\Re z\right)\cosh\left(\Im z\right)-i\sin\left(\Re z\right)\sinh\left(\Im z\right) \end{align*}(3)
\begin{align*} \tan z & =\tan\left(\Re z+i\Im z\right)\\ & =\frac{\sin\left(2\Re z\right)+\sin\left(2i\Im z\right)}{\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\\ & =\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)} \end{align*}(4)
\begin{align*} \sin^{-1}z & =\sin^{-1}\left(\Re z+i\Im z\right)\\ & =\frac{2\sin\left(\Re z-i\Im z\right)}{-\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\\ & =2\frac{\sin\left(\Re z\right)\cos\left(i\Im z\right)-\cos\left(\Re z\right)\sin\left(i\Im z\right)}{-\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\\ & =2\frac{\sin\left(\Re z\right)\cosh\left(\Im z\right)-i\cos\left(\Re z\right)\sinh\left(\Im z\right)}{-\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \end{align*}(5)
\begin{align*} \cos^{-1}z & =\cos^{-1}\left(\Re z+i\Im z\right)\\ & =\frac{2\cos\left(\Re z-i\Im z\right)}{\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\\ & =2\frac{\cos\left(\Re z\right)\cos\left(i\Im z\right)+\sin\left(\Re z\right)\sin\left(i\Im z\right)}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\\ & =2\frac{\cos\left(\Re z\right)\cosh\left(\Im z\right)+i\sin\left(\Re z\right)\sinh\left(\Im z\right)}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)} \end{align*}(6)
\begin{align*} \tan^{-1}z & =\tan^{-1}\left(\Re z+i\Im z\right)\\ & =\frac{\sin\left(2\Re z\right)-\sin\left(2i\Im z\right)}{-\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\\ & =\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)} \end{align*}双曲線関数の実部と虚部
(1)
\[ \sinh z=\sinh\left(\Re z\right)\cos\left(\Im z\right)+i\cosh\left(\Re z\right)\sin\left(\Im z\right) \](2)
\[ \cosh z=\cosh\left(\Re z\right)\cos\left(\Im z\right)+i\sinh\left(\Re z\right)\sin\left(\Im z\right) \](3)
\[ \tanh z=\frac{\sinh\left(2\Re z\right)+i\sin\left(2\Im z\right)}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)} \](4)
\[ \sinh^{-1}z=2\frac{\sinh\left(\Re z\right)\cos\left(\Im z\right)-i\cosh\left(\Re z\right)\sin\left(\Im z\right)}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)} \](5)
\[ \cosh^{-1}z=2\frac{\cosh\left(\Re z\right)\cos\left(\Im z\right)-i\sinh\left(\Re z\right)\sin\left(\Im z\right)}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)} \](6)
\[ \tanh^{-1}z=\frac{\sinh\left(2\Re z\right)-i\sin\left(2\Im z\right)}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)} \](1)
\begin{align*} \sinh z & =\sinh\left(\Re z+i\Im z\right)\\ & =\sinh\left(\Re z\right)\cosh\left(i\Im z\right)+\cosh\left(\Re z\right)\sinh\left(i\Im z\right)\\ & =\sinh\left(\Re z\right)\cos\left(\Im z\right)+i\cosh\left(\Re z\right)\sin\left(\Im z\right) \end{align*}(2)
\begin{align*} \cosh z & =\cosh\left(\Re z+i\Im z\right)\\ & =\cosh\left(\Re z\right)\cosh\left(i\Im z\right)+\sinh\left(\Re z\right)\sinh\left(i\Im z\right)\\ & =\cosh\left(\Re z\right)\cos\left(\Im z\right)+i\sinh\left(\Re z\right)\sin\left(\Im z\right) \end{align*}(3)
\begin{align*} \tanh z & =\tanh\left(\Re z+i\Im z\right)\\ & =\frac{\sinh\left(2\Re z\right)+\sinh\left(2i\Im z\right)}{\cosh\left(2\Re z\right)+\cosh\left(2i\Im z\right)}\\ & =\frac{\sinh\left(2\Re z\right)+i\sin\left(2\Im z\right)}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)} \end{align*}(4)
\begin{align*} \sinh^{-1}z & =\sinh^{-1}\left(\Re z+i\Im z\right)\\ & =\frac{2\sinh\left(\Re z-i\Im z\right)}{\cosh\left(2\Re z\right)-\cosh\left(2i\Im z\right)}\\ & =2\frac{\sinh\left(\Re z\right)\cosh\left(i\Im z\right)-\cosh\left(\Re z\right)\sinh\left(i\Im z\right)}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)}\\ & =2\frac{\sinh\left(\Re z\right)\cos\left(\Im z\right)-i\cosh\left(\Re z\right)\sin\left(\Im z\right)}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)} \end{align*}(5)
\begin{align*} \cosh^{-1}z & =\cosh^{-1}\left(\Re z+i\Im z\right)\\ & =\frac{2\cosh\left(\Re z-i\Im z\right)}{\cosh\left(2\Re z\right)+\cosh\left(2i\Im z\right)}\\ & =2\frac{\cosh\left(\Re z\right)\cosh\left(i\Im z\right)-\sinh\left(\Re z\right)\sinh\left(i\Im z\right)}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)}\\ & =2\frac{\cosh\left(\Re z\right)\cos\left(\Im z\right)-i\sinh\left(\Re z\right)\sin\left(\Im z\right)}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)} \end{align*}(6)
\begin{align*} \tanh^{-1}z & =\tanh^{-1}\left(\Re z+i\Im z\right)\\ & =\frac{\sinh\left(2\Re z\right)-\sinh\left(2i\Im z\right)}{\cosh\left(2\Re z\right)-\cosh\left(2i\Im z\right)}\\ & =\frac{\sinh\left(2\Re z\right)-i\sin\left(2\Im z\right)}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)} \end{align*}
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三角関数と双曲線関数の積分
\[
\int f(\cos x,\sin x)dx=\int f\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)\frac{2}{1+t^{2}}dt\cnd{t=\tan\frac{x}{2}}
\]
三角関数と双曲線関数のn乗積分
\[
\int\sin^{2n+m_{\pm}}xdx=\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}xdx\right\}
\]
正接関数・双曲線正接関数の多重対数関数表示
\[
\tan^{\pm1}z=i^{\pm1}\left(1+2\Li_{0}\left(\mp e^{2iz}\right)\right)
\]
1と3角関数・双曲線関数
\[
1+\sin z=\left(\cos\frac{z}{2}+\sin\frac{z}{2}\right)^{2}
\]