Trigonometrische Funktionen mit komplexen Werten

$$\sin(x+iy)=\sin x \cos iy + \cos x \sin iy = \sin x \cosh y + i \cos x \sinh y$$

$$\cos(x+iy))\cos x \cos iy - \sin x \sin iy = \cos x \cosh y - i \sin x \sinh y$$

$$\sin iz=i \sinh z \qquad \sinh iz = i \sin z$$

$$\cos iz= \cosh z \qquad \cosh iz = \cos z$$

$$\sin z = \frac{1}{2i}\left (e^{iz}-e^{-iz}\right ) \qquad \cos z = \frac{1}{2}\left ( e^{iz}+ e^{-iz} \right )$$

$$\sinh z = \frac{1}{2i}\left (e^{z}-e^{-z}\right ) \qquad \cosh z = \frac{1}{2}\left ( e^{z}+ e^{-z} \right )$$