三角関数・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)
\[ \tan z=\frac{1}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(\sin\left(2\Re z\right)+i\sinh\left(2\Im z\right)\right) \](4)
\[ \sin^{-1}z=\frac{2}{-\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(\sin\left(\Re z\right)\cosh\left(\Im z\right)-i\cos\left(\Re z\right)\sinh\left(\Im z\right)\right) \](5)
\[ \cos^{-1}z=\frac{2}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(\cos\left(\Re z\right)\cosh\left(\Im z\right)+i\sin\left(\Re z\right)\sinh\left(\Im z\right)\right) \](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\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\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 & =\frac{1}{\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\left(\sin\left(2\Re z\right)+\sin\left(2i\Im z\right)\right)\\ & =\frac{1}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(\sin\left(2\Re z\right)+i\sinh\left(2\Im z\right)\right) \end{align*}(4)
\begin{align*} \sin^{-1}z & =\frac{2\sin\overline{z}}{-\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\\ & =\frac{2}{-\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(\sin\left(\Re z\right)\cosh\left(\Im z\right)-i\cos\left(\Re z\right)\sinh\left(\Im z\right)\right) \end{align*}(5)
\begin{align*} \cos^{-1}z & =\frac{2\cos\overline{z}}{\cos\left(2\Re z\right)+\cos\left(2i\Im z\right)}\\ & =\frac{2}{\cos\left(2\Re z\right)+\cosh\left(2\Im z\right)}\left(\cos\left(\Re z\right)\cosh\left(\Im z\right)+i\sin\left(\Re z\right)\sinh\left(\Im z\right)\right) \end{align*}(6)
\begin{align*} \tan^{-1}z & =\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)
\[ \tanh z=\frac{1}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)}\left(\sinh\left(2\Re z\right)+i\sin\left(2\Im z\right)\right) \](4)
\[ \sinh^{-1}z=\frac{2}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)}\left(\sinh\left(\Re z\right)\cos\left(\Im z\right)-i\cosh\left(\Re z\right)\sin\left(\Im z\right)\right) \](5)
\[ \cosh^{-1}z=\frac{2}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)}\left(\cosh\left(\Re z\right)\cos\left(\Im z\right)-i\sinh\left(\Re z\right)\sin\left(\Im z\right)\right) \](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\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*}(1)-2
\begin{align*} \sinh z & =i^{-1}\sin\left(iz\right)\\ & =i^{-1}\left\{ \sin\left(\Re\left(iz\right)\right)\cosh\left(\Im\left(iz\right)\right)+i\cos\left(\Re\left(iz\right)\right)\sinh\left(\Im\left(iz\right)\right)\right\} \\ & =\cos\left(\Re\left(iz\right)\right)\sinh\left(\Im\left(iz\right)\right)-i\sin\left(\Re\left(iz\right)\right)\cosh\left(\Im\left(iz\right)\right)\\ & =\cos\left(-\Im z\right)\sinh\left(\Re z\right)-i\sin\left(-\Im z\right)\cosh\left(\Re z\right)\\ & =\cos\left(\Im z\right)\sinh\left(\Re z\right)+i\sin\left(\Im z\right)\cosh\left(\Re z\right) \end{align*}(2)
\begin{align*} \cosh z & =\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*}(2)-2
\begin{align*} \cosh z & =\cos\left(iz\right)\\ & =\cos\left(\Re\left(iz\right)\right)\cosh\left(\Im\left(iz\right)\right)-i\sin\left(\Re\left(iz\right)\right)\sinh\left(\Im\left(iz\right)\right)\\ & =\cos\left(-\Im z\right)\cosh\left(\Re z\right)-i\sin\left(-\Im z\right)\sinh\left(\Re z\right)\\ & =\cos\left(\Im z\right)\cosh\left(\Re z\right)+i\sin\left(\Im z\right)\sinh\left(\Re z\right) \end{align*}(3)
\begin{align*} \tanh z & =\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{1}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)}\left(\sinh\left(2\Re z\right)+i\sin\left(2\Im z\right)\right) \end{align*}(3)-2
\begin{align*} \tanh z & =i^{-1}\tan\left(iz\right)\\ & =i^{-1}\left\{ \frac{1}{\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\left(\sin\left(2\Re\left(iz\right)\right)+i\sinh\left(2\Im\left(iz\right)\right)\right)\right\} \\ & =\frac{1}{\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\left(\sinh\left(2\Im\left(iz\right)\right)-i\sin\left(2\Re\left(iz\right)\right)\right)\\ & =\frac{1}{\cos\left(-2\Im z\right)+\cosh\left(2\Re z\right)}\left(\sinh\left(2\Re z\right)-i\sin\left(-2\Im z\right)\right)\\ & =\frac{1}{\cos\left(2\Im z\right)+\cosh\left(2\Re z\right)}\left(\sinh\left(2\Re z\right)+i\sin\left(2\Im z\right)\right) \end{align*}(4)
\begin{align*} \sinh^{-1}z & =\frac{2\sinh\overline{z}}{\cosh\left(2\Re z\right)-\cosh\left(2i\Im z\right)}\\ & =\frac{2}{\cosh\left(2\Re z\right)-\cos\left(2\Im z\right)}\left(\sinh\left(\Re z\right)\cos\left(\Im z\right)-i\cosh\left(\Re z\right)\sin\left(\Im z\right)\right) \end{align*}(4)-2
\begin{align*} \sinh^{-1}z & =i\sin^{-1}\left(iz\right)\\ & =i\left\{ \frac{2}{-\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\left(\sin\left(\Re\left(iz\right)\right)\cosh\left(\Im\left(iz\right)\right)-i\cos\left(\Re\left(iz\right)\right)\sinh\left(\Im\left(iz\right)\right)\right)\right\} \\ & =\frac{2}{-\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\left(\cos\left(\Re\left(iz\right)\right)\sinh\left(\Im\left(iz\right)\right)+i\sin\left(\Re\left(iz\right)\right)\cosh\left(\Im\left(iz\right)\right)\right)\\ & =\frac{2}{-\cos\left(-2\Im z\right)+\cosh\left(2\Re z\right)}\left(\cos\left(-\Im z\right)\sinh\left(\Re z\right)+i\sin\left(-\Im z\right)\cosh\left(\Re z\right)\right)\\ & =\frac{2}{-\cos\left(2\Im z\right)+\cosh\left(2\Re z\right)}\left(\cos\left(\Im z\right)\sinh\left(\Re z\right)-i\sin\left(\Im z\right)\cosh\left(\Re z\right)\right) \end{align*}(5)
\begin{align*} \cosh^{-1}z & =\frac{2\cosh\overline{z}}{\cosh\left(2\Re z\right)+\cosh\left(2i\Im z\right)}\\ & =\frac{2}{\cosh\left(2\Re z\right)+\cos\left(2\Im z\right)}\left(\cosh\left(\Re z\right)\cos\left(\Im z\right)-i\sinh\left(\Re z\right)\sin\left(\Im z\right)\right) \end{align*}(5)-2
\begin{align*} \cosh^{-1}z & =\cos^{-1}\left(iz\right)\\ & =\frac{2}{\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\left(\cos\left(\Re\left(iz\right)\right)\cosh\left(\Im\left(iz\right)\right)+i\sin\left(\Re\left(iz\right)\right)\sinh\left(\Im\left(iz\right)\right)\right)\\ & =\frac{2}{\cos\left(-2\Im z\right)+\cosh\left(2\Re z\right)}\left(\cos\left(-\Im z\right)\cosh\left(\Re z\right)+i\sin\left(-\Im z\right)\sinh\left(\Re z\right)\right)\\ & =\frac{2}{\cos\left(2\Im z\right)+\cosh\left(2\Re z\right)}\left(\cos\left(\Im z\right)\cosh\left(\Re z\right)-i\sin\left(\Im z\right)\sinh\left(\Re z\right)\right) \end{align*}(6)
\begin{align*} \tanh^{-1}z & =\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*}(6)-2
\begin{align*} \tanh^{-1}z & =i\tan^{-1}\left(iz\right)\\ & =i\left\{ \frac{\sin\left(2\Re\left(iz\right)\right)-i\sinh\left(2\Im\left(iz\right)\right)}{-\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\right\} \\ & =\frac{\sinh\left(2\Im\left(iz\right)\right)+i\sin\left(2\Re\left(iz\right)\right)}{-\cos\left(2\Re\left(iz\right)\right)+\cosh\left(2\Im\left(iz\right)\right)}\\ & =\frac{\sinh\left(2\Re z\right)+i\sin\left(-2\Im z\right)}{-\cos\left(-2\Im z\right)+\cosh\left(2\Re z\right)}\\ & =\frac{\sinh\left(2\Re z\right)-i\sin\left(2\Im z\right)}{-\cos\left(2\Im z\right)+\cosh\left(2\Re z\right)} \end{align*}
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三角関数と双曲線関数の加法定理
\[
\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y
\]
オイラーの公式の応用
\[
\cos z\pm i\sin z=e^{\pm iz}
\]
巾関数と逆三角関数・逆双曲線関数の積の積分
\[
\int z^{\alpha}\Sin^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sin^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C
\]
逆三角関数と逆双曲線関数の関係
\[
\Sin^{\bullet}\left(iz\right)=i\Sinh^{\bullet}z
\]