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
Proofs: 1
Corollaries: 1 2 3
Definitions: 4 5 6
Lemmas: 7
Proofs: 8 9 10 11
Propositions: 12 13 14 15 16 17 18