---
title: "Show u = x²+2xy-y² harmonic; find v"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:harmonic-x2-2xy-y2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/harmonic-x2-2xy-y2
site: BCT Engineering
---

# Show u = x²+2xy-y² harmonic; find v

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2070 Bhadra, 5 marks
- Video: [Harmonic Conjugate Problem 1|Analytic Functions | Engineering Maths](https://www.youtube.com/watch?v=sPRnBaYK8JQ) — Alex Maths Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/harmonic-x2-2xy-y2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Show that u(x, y) = x² + 2xy - y² is a harmonic function and determine v(x, y) in such a way that f(z) = u(x, y) + iv(x, y) is analytic.

## Short answer

u is harmonic (u_xx+u_yy=0). v=2xy+y²−x²+ constant, giving f(z)=(1−i)z²+C.

## Answer

#### Show that u is harmonic
u = x² + 2xy − y².

```text
u_x = 2x + 2y     u_xx = 2
u_y = 2x − 2y     u_yy = −2
```

u_xx + u_yy = 0, and the second derivatives are continuous. So u is harmonic.

#### Find v (C–R equations: v_y = u_x, v_x = −u_y)
Integrate v_y = 2x + 2y with respect to y:

v = 2xy + y² + g(x)

Then v_x = 2y + g′(x) must equal −u_y = −2x + 2y. So g′(x) = −2x and g(x) = −x² + c.

**Answer: v = 2xy + y² − x² + c**

#### The analytic function
f′(z) = u_x(z, 0) − i u_y(z, 0) = 2z − 2iz = 2(1 − i)z

f(z) = (1 − i) z² + C.

Check: (1 − i)(x² − y² + 2ixy) = (x² − y² + 2xy) + i(2xy − x² + y²). The real part is u ✓ and the imaginary part is v ✓.

## Check yourself

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

1. Which shows u is harmonic?
   A. u_xx = 2 and u_yy = −2 sum to 0
   B. u is a polynomial
   C. u_x = u_y
   D. u_xy = 0
   Answer: A. Laplace's equation. (Key point: Harmonic)

2. The harmonic conjugate v is:
   A. 2xy + x² − y² + c
   B. x² + y²
   C. 2xy + y² − x² + c
   D. 2x + 2y
   Answer: C. v_y = 2x + 2y then g′ = −2x. (Key point: v)

3. f′(z) =
   A. 2(1−i) z
   B. 2(1+i) z
   C. 2z
   D. (1−i)
   Answer: A. u_x − i u_y = 2z − 2iz. (Key point: f′)

4. f(z) =
   A. (1−i) z + C
   B. z² + C
   C. (1−i) z² + C
   D. (1+i) z² + C
   Answer: C. Integrate. (Key point: f)

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