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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 \(1\)?

| | | | | created: 2016-02-29 22:58:10 | modified: 2020-06-14 16:39:05 | by: bookofproofs | references: [581]

1.Proof: (related to "Unit Circle")

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983