---
title: "Analytic function from u = x² - y²"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:analytic-function-x2-y2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/analytic-function-x2-y2
site: BCT Engineering
---

# Analytic function from u = x² - y²

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2069 Bhadra
- Video: [If u=x²-y² find the conjugate harmonix function- Milnes Thomson Method](https://www.youtube.com/watch?v=lavqyKt7NWg) — Btech Maths Hub
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/analytic-function-x2-y2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Determine the analytic function f(z) = u(x,y) + iv(x,y) if u(x,y) = x² - y².

## Short answer

u=x²−y² is harmonic; f(z)=z²+ic′ with v=2xy+constant.

## Answer

u = x² − y².

#### Check that u is harmonic
u_x = 2x, u_y = −2y, u_xx = 2, u_yy = −2, so u_xx + u_yy = 0 ✓.

#### Milne-Thomson method
f′(z) = u_x − i u_y. Put x = z, y = 0:

f′(z) = 2z − i·0 = 2z

Integrating: f(z) = z² + c.

#### Find v
z² = (x² − y²) + 2ixy. So v = 2xy + constant.

Check with C–R: v_y = 2x = u_x ✓ and v_x = 2y = −u_y ✓.

**Answer: f(z) = z² + ic′ (that is, v = 2xy + c′).**

## Check yourself

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

1. u = x² − y² is harmonic because:
   A. u is linear
   B. u_xx + u_yy = 2 − 2 = 0
   C. u_xx = u_yy
   D. u_x = 0
   Answer: B. Laplace. (Key point: Harmonic)

2. f′(z) with x=z, y=0 is:
   A. z²
   B. z
   C. 2z
   D. 2
   Answer: C. u_x = 2x. (Key point: f′)

3. f(z) =
   A. z³
   B. 2z
   C. z + c
   D. z² + c
   Answer: D. Integrate 2z. (Key point: Result)

4. The imaginary part v is:
   A. x² − y²
   B. 2xy + const
   C. x² + y²
   D. xy
   Answer: B. Im z² = 2xy. (Key point: v)

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