---
title: "Harmonic function and conjugate"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:harmonic-function-and-conjugate-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/harmonic-function-and-conjugate-two
site: BCT Engineering
---

# Harmonic function and conjugate

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2082 Kartik (new course), 2 marks
- Video: [Harmonic Functions & Conjugates Explained | Complex Analysis #3](https://www.youtube.com/watch?v=Vhm225csPsY) — TheMathCoach
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/harmonic-function-and-conjugate-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Show that the function u(x, y) = 2x + x³ − 3xy² is harmonic.

## Short answer

u_xx = 6x, u_yy = -6x, so u_xx+u_yy=0. Hence u is harmonic (Laplace equation satisfied). It is Re(2z+z^3).

## Answer

A function is harmonic if it satisfies Laplace's equation u_xx + u_yy = 0.

```text
u = 2x + x³ − 3xy²
u_x = 2 + 3x² − 3y²     u_xx = 6x
u_y = −6xy              u_yy = −6x
u_xx + u_yy = 6x − 6x = 0
```

All second partial derivatives are continuous, so **u is harmonic**. (u = Re(2z + z³).)

## Check yourself

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

1. Harmonic means:
   A. u_x = u_y
   B. u_xy = 0
   C. u_xx + u_yy = 0
   D. u_xx = u_yy
   Answer: C. Laplace's equation. (Key point: Test)

2. For u = 2x + x^3 - 3xy^2, u_xx =
   A. 3x^2
   B. 6x - 6y
   C. 2 + 6x
   D. 6x
   Answer: D. u_x = 2+3x^2-3y^2. (Key point: u_xx)

3. u_yy =
   A. -6x
   B. 6x
   C. -6y
   D. -6xy
   Answer: A. u_y = -6xy, u_yy = -6x; sum is 0. (Key point: u_yy)

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