---
title: "Complex Differentiability and the Cauchy-Riemann Equations"
description: "Define the complex derivative as a limit over all approach directions, derive the Cauchy-Riemann equations, and state the continuity condition that makes them sufficient."
canonical: https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss
source: https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss.md
retrieved: 2026-11-12
---

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# Complex Differentiability and the Cauchy-Riemann Equations

Define the complex derivative as a limit over all approach directions, derive the Cauchy-Riemann equations, and state the continuity condition that makes them sufficient.

Subject: Mathematics · Area: Complex Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss

## Ready when they can

- Define the complex difference quotient, derive the Cauchy-Riemann equations, and recover the derivative when differentiability is established
- Distinguish differentiability sets from holomorphic neighbourhoods, including real-smooth failures
- State the sufficient continuity hypothesis precisely and show counterexamples where CR holds without differentiability

## Lesson: One derivative from every direction

The complex derivative of f at z0 is the limit of [f(z0 + h) - f(z0)] / h as h tends to 0 through nonzero complex values, with f defined on an open neighbourhood of z0. The limit must be the same for every way h can approach 0, not just along the real or imaginary axis. This all-path demand is what makes complex differentiability stronger than real differentiability of the two component functions.

**Example.** Derive the Cauchy-Riemann equations. Write f = u + iv and let h = t be real. The quotient tends to u_x + i v_x. Now let h = ik be imaginary. The quotient tends to (u_y + i v_y) / i, which is v_y - i u_y. Equating real and imaginary parts gives u_x = v_y and u_y = -v_x. Once differentiability is established, f'(z0) = u_x + i v_x.

The equations are necessary but not sufficient by themselves. The sufficient hypothesis adds that the four partial derivatives exist in a neighbourhood of z0 and are continuous at z0, together with the equations holding at z0. Failure of the equations proves non-differentiability. Failure of continuity alone does not prove non-differentiability; a direct difference quotient can still settle the question.

**Example.** For z squared, u = x² - y² and v = 2xy. The four partials are u_x = 2x, u_y = -2y, v_x = 2y, v_y = 2x. Both equations hold everywhere. Polynomial partials are continuous, so f'(z) = 2x + i 2y = 2z.

**Example.** For f(z) = conj(z)²/z for z ≠ 0 and f(0) = 0, real increments give f(t) = t, so u_x = 1 and v_x = 0. Imaginary increments give f(it) = it, so u_y = 0 and v_y = 1. Both CR equations hold: u_x = v_y = 1 and u_y = -v_x = 0. The quotient Q(h) = (conj(h)/h)² equals 1 on real h and -1 on h = t(1+i). Two paths disagree, so no derivative at 0. Also |f(z)| = |z|, so f is continuous at 0 despite failing differentiability.

An open neighbourhood of a point contains a disk of positive radius around it. A function is holomorphic on an open set if differentiable at every point of that set. Holomorphic at a point means differentiable throughout some open neighbourhood of it. The conjugate function has u = x and v = -y, so u_x = 1 and v_y = -1 everywhere: nowhere differentiable, hence nowhere holomorphic.

**Example.** The function |z| squared satisfies the equations only at 0. At 0 the quotient |h| squared over h has modulus |h|, which tends to 0, so the derivative is 0 at 0 only. It is nowhere holomorphic.

**Recap.** Define the derivative over all paths, derive the equations by equating axial approaches, add continuous partials for sufficiency, and distinguish pointwise from holomorphic.

## Practice

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

## Needs first

- [Complex Functions and the Complex Plane as a Domain](https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs)
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9)

## Opens up

- [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P)
- [Harmonic Functions and What Analyticity Forces](https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4)
- [Conformal Maps and Mobius Transformations](https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI)