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

# Harmonic function and conjugate

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

- Asked in: 2081 Chaitra (new course), 2+2 marks
- Video: [Harmonic Functions and Harmonic Conjugates | Complex Variables](https://www.youtube.com/watch?v=7XHah5XF7L4) — Faculty of Khan
- 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Prove that u = y³ − 3x²y is a harmonic function. Find its harmonic conjugate and hence construct the corresponding analytic function w = f(z).

## Short answer

u satisfies Laplace's equation. Harmonic conjugate v = x³ − 3xy² + c, analytic function f(z) = i z³ + c'.

## Answer

#### Harmonic
```text
u = y³ − 3x²y
u_x = −6xy     u_xx = −6y
u_y = 3y² − 3x²   u_yy = 6y
u_xx + u_yy = 0
```
So u satisfies Laplace's equation and is harmonic.

#### Harmonic conjugate v
Use the Cauchy–Riemann equations v_y = u_x and v_x = −u_y.

```text
v_y = −6xy   → v = −3xy² + g(x)
v_x = −3y² + g'(x) = −u_y = 3x² − 3y²
g'(x) = 3x²  → g = x³
v = x³ − 3xy² + c
```

#### Analytic function
```text
f(z) = u + iv = (y³ − 3x²y) + i (x³ − 3xy²) + ic
```

Since i z³ = i(x³ + 3ix²y − 3xy² − iy³) = (y³ − 3x²y) + i(x³ − 3xy²), we get

**Answer: v = x³ − 3xy² + c and w = f(z) = i z³ + c'.**

## Check yourself

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

1. u = y^3-3x^2y is harmonic because:
   A. u_x = u_y
   B. u_xx = 0
   C. u_xx + u_yy = -6y + 6y = 0
   D. u_xy = 0
   Answer: C. Laplace's equation holds. (Key point: Harmonic)

2. Conjugate v uses:
   A. v_y = u_y
   B. v_x = u_x
   C. v_x = u_y
   D. v_y = u_x and v_x = -u_y
   Answer: D. Cauchy-Riemann. (Key point: CR eq)

3. v is:
   A. x^3 - 3xy^2 + c
   B. 3xy^2 - x^3
   C. y^3 - 3x^2y
   D. x^3 + 3xy^2
   Answer: A. Integrate v_y=-6xy, g'=3x^2. (Key point: v)

4. f(z) is:
   A. z^3
   B. i z^3 + c
   C. -i z^3
   D. i z^2
   Answer: B. i z^3 = (y^3-3x^2y)+i(x^3-3xy^2). (Key point: f(z))

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