exp z = 1 iff z is an integer multiple of 2πi

Builds on Complex exponential Euler's formula Integers

Statement

Let $z \in \mathbb{C}$ . Then $e^z = 1$ if and only if $z = 2 \pi i k$ for some integer $k$ . Here $e^z$ is the complex exponential of $z$ .

Remarks

Proof: write $z = x + iy$ with $x, y$ real. Then $e^z = e^x (\cos y + i \sin y)$ . If $e^z = 1$ , then $e^x = |e^z| = 1$ , so $x = 0$ . Then $\cos y = 1$ and $\sin y = 0$ , so $y = 2 \pi k$ . The converse follows from Euler's formula. It follows that $e^z = e^w$ if and only if $z - w = 2 \pi i k$ for some $k \in \mathbb{Z}$ . This is the reason why the complex logarithm has many values.

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Complex numbers object
Integers object
Complex exponential definition
Euler's formula statement
exp z = 1 iff z is an integer multiple of 2πi statement

Tag	`013AHC4DRT`
URI	`m2://013AHC4DRT`
Aliases	`kernel-of-the-exponential`
Kind	statement, proposition
Writing	W0 stub
MSC	33B10
Status	This copy of the site has no build data. Run `m2 status 013AHC4DRT`.
Uses	Complex exponential, Euler's formula, Integers

exp z = 1 iff z is an integer multiple of 2πi

Statement

Remarks

Reading path

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