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

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