Welcome guest
You're not logged in.
287 users online, thereof 0 logged in

Proposition: Complex Exponential Function

The complex exponential series


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”.



| | | | | 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

Edit or AddNotationAxiomatic Method

This work was contributed under CC BY-SA 4.0 by:

This work is a derivative of:

Bibliography (further reading)