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Lemma: Unit Circle

Let \(x\in\mathbb R\) be any real number and let \(z\) be the complex number, for which \(x\) is the imaginary part, i.e. \(x=\Im(z)\Longleftrightarrow z:=ix\).

The distance of the complex exponential function from the point of origin is equal \(1\), formally

\[|\exp(ix)|=1\quad\quad\text{for all }x\in\mathbb R.\]

Geometrically, the complex numbers \(\exp(ix)\) form a figure called the unit circle:

Fun questions

• For which values of \(x\) does \(\exp(ix)\) reach the complex number \(i\)?
• For which values of \(x\) does \(\exp(ix)\) reach the complex number \(1\)?

Table of Contents

Proofs: 1

Mentioned in:

Definitions: 1 2
Lemmas: 3
Proofs: 4 5 6
Propositions: 7

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983