---
title: "Derive Cauchy-Riemann equations (Cartesian)"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:cauchy-riemann-cartesian"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/cauchy-riemann-cartesian
site: BCT Engineering
---

# Derive Cauchy-Riemann equations (Cartesian)

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2074 Bhadra, 1+4 marks; 2073 Bhadra, 5 marks (asked 2 times)
- Video: [The Cauchy-Riemann Equations: a proof](https://www.youtube.com/watch?v=NuW59gIy2MI) — Michael Barrus
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/cauchy-riemann-cartesian
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define an analytic function of a complex variable. Derive the Cauchy-Riemann equations in Cartesian form for f(z) = u + iv.

## Short answer

An analytic function is differentiable in a neighborhood of each point. Differentiating f(z)=u+iv along the real and imaginary axes and equating the two expressions for f′(z) gives u_x=v_y and u_y=−v_x, the Cauchy–Riemann equations in Cartesian form.

## Answer

#### Analytic function
A function f(z) is **analytic** at a point z₀ if it is differentiable at z₀ and at every point in some neighbourhood of z₀. It is analytic in a region if it is differentiable at every point of that region.

#### Cauchy–Riemann equations
**Theorem.** If f(z) = u(x, y) + i v(x, y) is analytic in a region, then at every point of it

∂u/∂x = ∂v/∂y  and  ∂u/∂y = −∂v/∂x.

#### Derivation
Since f is differentiable at z = x + iy, the limit

f′(z) = lim_{δz→0} [f(z + δz) − f(z)]/δz

exists and is the same along every path by which δz → 0. Here δz = δx + i δy.

**Path 1: along the real axis** (δy = 0, so δz = δx → 0):

f′(z) = lim [u(x + δx, y) − u(x, y)]/δx + i [v(x + δx, y) − v(x, y)]/δx = ∂u/∂x + i ∂v/∂x.  ... (1)

**Path 2: along the imaginary axis** (δx = 0, so δz = i δy → 0):

f′(z) = lim [u(x, y + δy) − u(x, y)]/(i δy) + i [v(x, y + δy) − v(x, y)]/(i δy)
     = (1/i) ∂u/∂y + ∂v/∂y = ∂v/∂y − i ∂u/∂y.  ... (2)

The two values must be equal. Equate real and imaginary parts of (1) and (2):

```text
∂u/∂x = ∂v/∂y
∂v/∂x = −∂u/∂y
```

These are the **Cauchy–Riemann (C–R) equations**. ∎

Also f′(z) = u_x + i v_x = v_y − i u_y.

**Sufficient condition.** If u, v, and their first partial derivatives are continuous in a region and satisfy the C–R equations there, then f(z) = u + iv is analytic in that region.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A function is analytic at z0 if:
   A. It is differentiable at z0 and in a neighbourhood of z0
   B. It is continuous at z0
   C. It is real valued
   D. It is differentiable only at z0
   Answer: A. Analyticity needs differentiability throughout a neighbourhood. (Key point: Analytic)

2. The C–R derivation relies on:
   A. The function being real
   B. The derivative limit being the same along every path
   C. Continuity alone
   D. Integration over a contour
   Answer: B. Taking different paths gives two expressions that must be equal. (Key point: Idea)

3. Along the real axis (δy = 0), f′(z) equals:
   A. u_x − i v_x
   B. u_x + i v_x
   C. v_y − i u_y
   D. u_y + i v_y
   Answer: B. Only x changes. (Key point: Real axis path)

4. The C–R equations are:
   A. u_x = −v_y and u_y = v_x
   B. u_x = v_x and u_y = v_y
   C. u_x = v_y and u_y = −v_x
   D. u_x = u_y and v_x = v_y
   Answer: C. Equate real and imaginary parts of u_x + i v_x = v_y − i u_y. (Key point: Result)

## Related in Applied Mathematics

- [Cauchy-Riemann equations in polar form](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acauchy-riemann-polar) — Applied Mathematics, chapter 1, asked 2×
- [Analytic function from u = y³ - 3x²y](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aanalytic-function-y3-3x2y) — Applied Mathematics, chapter 1
- [Analytic function from u = log√(x²+y²)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aanalytic-function-log-sqrt) — Applied Mathematics, chapter 1, asked 2×
- [Analytic function from u = 3x²y - y³](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aanalytic-function-3x2y-y3) — Applied Mathematics, chapter 1
- [Analytic function from u = x² - y²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aanalytic-function-x2-y2) — Applied Mathematics, chapter 1

## Related in other subjects

- [Cauchy's formula and equilibrium equations](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Acauchy-formula-equilibrium) — Mechanics of Solids, chapter 2