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

# Harmonic function and conjugate

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

- Asked in: 2082 Chaitra (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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Define harmonic function. Show that the function u(x, y) = eˣ cos y is harmonic.

## Short answer

A function is harmonic if it satisfies Laplace's equation ∂²u/∂x²+∂²u/∂y²=0. For u = eˣ cos y, ∂²u/∂x² = eˣ cos y, ∂²u/∂y² = −eˣ cos y. Their sum is 0, so u is harmonic.

## Answer

#### Definition
A real function u(x,y) with continuous second partial derivatives is **harmonic** in a region if it satisfies Laplace's equation there:

∂²u/∂x² + ∂²u/∂y² = 0.

(The real and imaginary parts of any analytic function are harmonic.)

#### u = eˣ cos y
```text
∂u/∂x = eˣ cos y      ∂²u/∂x² = eˣ cos y
∂u/∂y = −eˣ sin y     ∂²u/∂y² = −eˣ cos y

∂²u/∂x² + ∂²u/∂y² = eˣ cos y − eˣ cos y = 0
```

So u = eˣ cos y is harmonic. It is the real part of the analytic function f(z) = eᶻ = eˣ(cos y + i sin y).

## Check yourself

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

1. u is harmonic if:
   A. u_x + u_y = 0
   B. u_xx = u_yy
   C. It has continuous second partials and u_xx + u_yy = 0
   D. u is a polynomial
   Answer: C. This is Laplace's equation. (Key point: Definition)

2. For u = e^x cos y, u_xx is:
   A. -e^x cos y
   B. e^x sin y
   C. -e^x sin y
   D. e^x cos y
   Answer: D. Differentiating e^x twice gives e^x. (Key point: u_xx)

3. u_yy equals:
   A. -e^x cos y
   B. e^x cos y
   C. e^x sin y
   D. -e^x sin y
   Answer: A. d/dy(-e^x sin y) = -e^x cos y; sum is zero. (Key point: u_yy)

## Related in Mathematics III

- [Harmonic function and conjugate](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aharmonic-function-and-conjugate-three) — Mathematics III, chapter 2
- [Harmonic function and conjugate](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aharmonic-function-and-conjugate-two) — Mathematics III, chapter 2

## Related in other subjects

- [Show u = ½log(x²+y²) harmonic; conjugate](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aharmonic-log-conjugate) — Applied Mathematics, chapter 1
- [Harmonic function; is arg(z) harmonic](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aharmonic-arg-conjugate) — Applied Mathematics, chapter 1
- [Harmonic exponentials; CTFS conjugate symmetry](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aharmonic-exponentials-conjugation) — Signals and Systems, chapter 2
- [Gamma function: ∫ sin⁶(x/2) cos⁶(x/2) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Agamma-sin6-cos6-half-x) — Mathematics I, chapter 2, asked 3×