Complex Functions

Key idea

Functions of a complex variable f(z).

In depth

A complex function f(z) takes complex inputs and gives complex outputs, visualised as a map of the plane. Writing f = u + iv separates real and imaginary parts. Many real functions extend naturally to the complex plane.

Worked example

f(z) = z² maps z = x + iy to (x² − y²) + 2xyi.

Real-world uses

• Describing two-dimensional flows, electric potentials and signals in engineering

Common mistake

Treating z as a vector and ignoring the complex structure.

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

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.