Complex Exponential and Logarithm

Key idea

Multi-valued log in the complex plane.

In depth

e^z = eˣ(cos y + i sin y). It is periodic with period 2πi. So the logarithm is multi-valued: log z = ln|z| + i(arg z + 2πk). Choosing a branch (principal value) makes it a function.

Worked example

e^(iπ) = −1, so log(−1) = iπ (principal value).

Formula

Real-world uses

• Describing rotating and decaying signals in physics and electrical engineering

Common mistake

Treating log as single-valued over the whole plane.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

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• Density of Rationals – Between any two reals there is a rational.
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• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.