---
title: "Cauchy-Riemann equations in polar form"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:cauchy-riemann-polar"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/cauchy-riemann-polar
site: BCT Engineering
---

# Cauchy-Riemann equations in polar form

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2073 Magh, 5 marks; 2072 Asoj, 5 marks (asked 2 times)
- Video: [The Cauchy Riemann Equations in Polar Coordinates](https://www.youtube.com/watch?v=atEAC_ASsYA) — Mike, the Mathematician
- 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-polar
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define an analytic function. Express the Cauchy-Riemann equations u_x = v_y and u_y = -v_x in polar form.

## Short answer

An analytic function f(z) is differentiable at a point and in a neighbourhood of that point. The Cauchy-Riemann equations in polar form, with f(z)=u(r,θ)+iv(r,θ), are ∂u/∂r = (1/r)∂v/∂θ and ∂v/∂r = −(1/r)∂u/∂θ.

## Answer

#### Analytic function
f(z) is **analytic** at z₀ if it is differentiable at z₀ and at every point of some neighbourhood of z₀.

#### C–R equations in polar form
Let z = re^{iθ} and f(z) = u(r, θ) + i v(r, θ). Then the C–R equations are

**∂u/∂r = (1/r) ∂v/∂θ  and  ∂v/∂r = −(1/r) ∂u/∂θ.**

#### Derivation
Use f′(z) = lim [f(z + δz) − f(z)]/δz and approach z in two ways. Here z = re^{iθ}.

**Radial direction** (θ fixed, r changes by δr): δz = e^{iθ} δr, so

f′(z) = (∂f/∂r)/e^{iθ} = e^{−iθ} (u_r + i v_r).  ... (1)

**Angular direction** (r fixed, θ changes by δθ): δz = i r e^{iθ} δθ, so

f′(z) = (∂f/∂θ)/(i r e^{iθ}) = (e^{−iθ}/(ir)) (u_θ + i v_θ) = (e^{−iθ}/r)(v_θ − i u_θ).  ... (2)

Equate (1) and (2) and cancel e^{−iθ}:

u_r + i v_r = (1/r) v_θ − i (1/r) u_θ

Equating real and imaginary parts:

```text
u_r = (1/r) v_θ
v_r = −(1/r) u_θ
```

#### Link with the Cartesian form
Using x = r cos θ, y = r sin θ and u_x = v_y, u_y = −v_x:

- v_θ = −r sin θ · v_x + r cos θ · v_y = r (u_y sin θ + u_x cos θ) = r u_r
- u_θ = −r sin θ · u_x + r cos θ · u_y = −r (v_y sin θ + v_x cos θ) = −r v_r

This gives the same polar equations. Also f′(z) = e^{−iθ}(u_r + i v_r).

## Check yourself

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

1. The polar form of the C–R equations is:
   A. u_r = (1/r) v_θ, v_r = −(1/r) u_θ
   B. u_r = r v_θ, v_r = −r u_θ
   C. u_r = −(1/r) v_θ, v_r = (1/r) u_θ
   D. u_r = v_θ, v_r = −u_θ
   Answer: A. The 1/r comes from δz = i r e^{iθ} δθ. (Key point: Polar C–R)

2. Along the radial direction δz equals:
   A. e^{iθ} δr
   B. r δθ
   C. e^{iθ} δθ
   D. i r e^{iθ} δr
   Answer: A. θ fixed, r changes. (Key point: Radial direction)

3. Along the angular direction, δz equals:
   A. i e^{iθ} δr
   B. r e^{iθ} δθ
   C. e^{iθ} δθ
   D. i r e^{iθ} δθ
   Answer: D. z = re^{iθ}, dz/dθ = i r e^{iθ}. (Key point: Angular direction)

4. The derivative in polar form is f′(z) =
   A. e^{iθ}(u_r + i v_r)
   B. e^{−iθ}(u_θ + i v_θ)
   C. e^{−iθ}(u_r + i v_r)
   D. (u_r + i v_r)/r
   Answer: C. From the radial path, f′ = (∂f/∂r)/e^{iθ}. (Key point: Derivative)

## Related in Applied Mathematics

- [Derive Cauchy-Riemann equations (Cartesian)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acauchy-riemann-cartesian) — Applied Mathematics, chapter 1, asked 2×
- [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 = 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 = 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
- [Laplace equation in polar form](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaplace-eq-polar-form) — Applied Mathematics, chapter 3, asked 2×

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