Complex exponential function ($e^z$)

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

The complex complex exponential function has nonzero derivative and maps the strip onto the annulus, with fibers given by translations by $2\pi i$. It is locally conformal, but not injective. No biholomorphism from the full strip onto an annulus exists because the strip is simply connected space and the annulus has a loop of nonzero winding number. The complex exponential function is a covering map realizing that topological distinction.

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.