---
title: "Complex Functions and the Complex Plane as a Domain"
description: "Treat a function of a complex variable as a map of the plane to itself, and set up limits and continuity in that setting."
canonical: https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs
source: https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs.md
retrieved: 2026-09-12
---

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# Complex Functions and the Complex Plane as a Domain

Treat a function of a complex variable as a map of the plane to itself, and set up limits and continuity in that setting.

Subject: Mathematics · Area: Complex Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs

## Ready when they can

- Describe the image of a line or circle under a simple complex map
- Define a limit of a complex function and explain why approach from every direction is required
- Split a complex function into its real and imaginary parts as two real functions of two variables

## Lesson: Functions that remap the whole plane

A complex function eats points and spits points, remapping the plane to itself. Feeding the real axis into w is z plus i lifts it to the line Im(w) is 1. Feeding the unit circle into w is 2z doubles it to the circle of radius 2. Squaring sends 1 plus i to 2i, while positive reals square to positive reals, so that ray maps into itself.

Limits in the plane are stricter than on the line. Approaching along the real axis is one direction among infinitely many, and a complex limit must survive them all. Functions can match on both axes yet split on the diagonal. Directional agreement is necessary but never sufficient, and this strictness motivates everything next.

**Example.** Splitting into u and v turns one complex map into two real surfaces. For w is z squared, u is x squared minus y squared and v is 2xy. Point checks anchor the abstraction: u(1, 2) is negative 3 and v(1, 2) is 4. Polar form predicts the same images faster, since squaring doubles angles and squares radii.

**Tip.** Sketch the input, track a few landmark points, and the image curve emerges. Compute first and theorise second. When Cartesian expansion and polar prediction agree, for example both giving 2i for 1 plus i, that agreement is the signal of mastery.

**Recap.** Track where points go, split the map into u and v, and demand agreement from every direction.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550)
- [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8)
- [The Argand Plane, Modulus and Argument](https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e)

## Opens up

- [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss)