---
title: "Harmonic function; is arg(z) harmonic"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:harmonic-arg-conjugate"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/harmonic-arg-conjugate
site: BCT Engineering
---

# Harmonic function; is arg(z) harmonic

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2075 Baisakh, 1+1+3 marks
- Video: [How to find a Harmonic Conjugate Complex Analysis](https://www.youtube.com/watch?v=tWX8YwKfd_k) — The Math Sorcerer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/harmonic-arg-conjugate
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define harmonic function. Is V = arg(z) harmonic? If yes, find a corresponding harmonic conjugate.

## Short answer

A harmonic function satisfies Laplace's equation u_xx+u_yy=0. V=arg(z) has V_xx+V_yy=0, so it is harmonic. A harmonic conjugate is U = −½ log(x²+y²) = −ln|z| (plus constant), making f = V + iU = −i log z analytic.

## Answer

#### Harmonic function
A real function φ(x, y) is **harmonic** in a region if it has continuous second-order partial derivatives there and satisfies Laplace's equation

∂²φ/∂x² + ∂²φ/∂y² = 0.

#### Is V = arg z harmonic?
V = arg z = tan⁻¹(y/x) (defined for z ≠ 0, on a branch).

```text
V_x = −y/(x² + y²)           V_y = x/(x² + y²)
V_xx = 2xy/(x² + y²)²        V_yy = −2xy/(x² + y²)²
```

V_xx + V_yy = 0, so **yes, V = arg z is harmonic** (in any region not containing the origin and not crossing the branch cut).

#### Harmonic conjugate
We want U such that f = V + iU is analytic. The C–R equations require V_x = U_y and V_y = −U_x.

```text
U_y = V_x = −y/(x² + y²)   →  U = −½ log(x² + y²) + g(x)
U_x = −V_y = −x/(x² + y²)  →  g′(x) = 0
```

So U = −½ log(x² + y²) = −log|z| (plus a constant).

Check: V + iU = θ − i ln r = −i(ln r + iθ) = −i log z, which is analytic.

**Answer: Yes, harmonic. The conjugate is U = −log√(x² + y²) = −ln r (+ constant).**

(If V is instead taken as the imaginary part of an analytic function, the function is log z = ln r + iθ, and the corresponding real part is u = ln r.)

## Check yourself

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

1. A function is harmonic if it satisfies:
   A. Laplace's equation φxx + φyy = 0
   B. φ = const
   C. φx = φy
   D. φxy = 0
   Answer: A. Plus continuous second derivatives. (Key point: Harmonic)

2. Is arg z harmonic?
   A. Only on the real axis
   B. Yes, V_xx + V_yy = 0 away from the origin
   C. No, since V_xx ≠ 0
   D. No, it is not defined
   Answer: B. V_xx = 2xy/(x²+y²)² and V_yy = −2xy/(x²+y²)². (Key point: Test)

3. To find U with V + iU analytic, we use:
   A. U = V
   B. V_x = −U_y
   C. V_x = U_x and V_y = U_y
   D. V_x = U_y and V_y = −U_x
   Answer: D. C–R with V as the real part. (Key point: Conjugate eq)

4. The harmonic conjugate is:
   A. −log√(x²+y²)
   B. log√(x²+y²)
   C. x²+y²
   D. tan⁻¹(y/x)
   Answer: A. V + iU = −i log z. (Key point: Result)

## Related in Applied Mathematics

- [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
- [Show u = x²+2xy-y² harmonic; find v](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aharmonic-x2-2xy-y2) — Applied Mathematics, chapter 1
- [Analytic function from u = y³ - 3x²y](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aanalytic-function-y3-3x2y) — Applied Mathematics, chapter 1

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