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".



Proofs: 1

  1. Proposition: Estimate for the Remainder Term of Complex Exponential Function
  2. Proposition: Functional Equation of the Complex Exponential Function
  3. Proposition: \(\exp(0)=1\) (Complex Case)
  4. Proposition: Continuity of Complex Exponential Function
  5. Proposition: Complex Conjugate of Complex Exponential Function

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

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