Complex exponential function ($e^z$)

The complex exponential is the entire functionIt maps $\mathbb{C}$ onto ${\mathbb{C}}^{\ast }$ and has kernel $2\pi i\mathbb{Z}$.

For real $x,y$, the complex exponential function has $e^{x+iy}=e^x(\cos y+i\sin y)$. Since the trigonometric factor lies on the unit circle,Thus the real part of the exponent controls its modulus, while the imaginary part controls its direction on the complex plane.

The complex sine is the entire functionFor real $x,y$, $\sin (x+iy)=\sin x\cosh y+i\cos x\sinh y$.

The kernel $2\pi i\mathbb{Z}$ of the complex exponential function givesThe same period appears as the translation generated by the monodromy of the complex inverse sine.