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

1. Proof : (related to "Complex Exponential Function")

2. Proposition : Estimate for the Remainder Term of Complex Exponential Function

3. Proposition : Functional Equation of the Complex Exponential Function

4. Proposition : \(\exp(0)=1\) (Complex Case)

5. Proposition : Continuity of Complex Exponential Function

6. Proposition : Complex Conjugate of Complex Exponential Function