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## Proposition: Complex Exponential Function

The complex exponential series

$\sum_{n=0}^\infty\frac{z^n}{n!}$

is an absolutely convergent complex series for every complex number $$z\in\mathbb C$$. It defines a function $$\exp:\mathbb C\mapsto \mathbb C$$, called the complex exponential function for all $$z\in\mathbb C$$.

$\exp(x):=\sum_{n=0}^\infty\frac{z^n}{n!},\quad\quad z\in\mathbb R.$

In the following interactive figure, in the “$$z$$-plane”, you can drag the circle’s midpoint, drag the segment’s endpoints or change the radius of the circle and get a feeling of how the complex exponential function deforms their images in the “$$w$$-plane”.

$$z$$-Plane

$$w$$-Plane

| | | | | created: 2014-02-21 21:02:24 | modified: 2018-09-08 16:52:24 | by: bookofproofs