---
title: "Show u = ½log(x²+y²) harmonic; conjugate"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:harmonic-log-conjugate"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/harmonic-log-conjugate
site: BCT Engineering
---

# Show u = ½log(x²+y²) harmonic; conjugate

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2068 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/harmonic-log-conjugate
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define harmonic function and show that the function u = ½ log(x² + y²) is harmonic and hence determine its harmonic conjugate function.

## Short answer

A harmonic function satisfies Laplace's equation. u=½ log(x²+y²) has u_xx+u_yy=0, hence harmonic. Its conjugate v satisfies v_y = u_x = x/(x²+y²), giving v = tan⁻¹(y/x) + c, so f(z)=u+iv = log z (plus constant).

## Answer

#### Harmonic function
A real function u(x, y) is **harmonic** in a region if it has continuous second partial derivatives and satisfies Laplace's equation u_xx + u_yy = 0 there.

#### Show that u = ½ log(x² + y²) is harmonic
```text
u_x  = x/(x² + y²)
u_y  = y/(x² + y²)
u_xx = [(x² + y²) − 2x²]/(x² + y²)² = (y² − x²)/(x² + y²)²
u_yy = [(x² + y²) − 2y²]/(x² + y²)² = (x² − y²)/(x² + y²)²
```

u_xx + u_yy = 0 for (x, y) ≠ (0, 0). Hence u is harmonic.

#### Harmonic conjugate v
By the C–R equations v_x = −u_y and v_y = u_x.

```text
v_y = x/(x² + y²)
Integrate w.r.t. y:   v = tan⁻¹(y/x) + g(x)
Differentiate w.r.t. x: v_x = −y/(x² + y²) + g′(x)
```

This must equal −u_y = −y/(x² + y²), so g′(x) = 0 and g is a constant.

**Answer: v = tan⁻¹(y/x) + c** (that is, the argument θ).

The analytic function is f(z) = u + iv = ln r + iθ = log z.

## Check yourself

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

1. u_xx for u = ½ log(x²+y²) is:
   A. (y² − x²)/(x²+y²)²
   B. 2x/(x²+y²)
   C. (x² − y²)/(x²+y²)²
   D. 1/(x²+y²)
   Answer: A. Differentiate x/(x²+y²). (Key point: Laplace)

2. u_xx + u_yy = 0 holds:
   A. For (x,y) ≠ (0,0)
   B. Only at the origin
   C. Only on axes
   D. Nowhere
   Answer: A. The terms cancel except at the origin. (Key point: Harmonic check)

3. The conjugate v satisfies:
   A. v_y = u_y
   B. v_y = u_x, v_x = −u_y
   C. v_y = −u_x, v_x = u_y
   D. v_x = u_x
   Answer: B. Standard C–R. (Key point: C–R)

4. The conjugate and f(z) are:
   A. v = tan⁻¹(y/x)+c, f = log z
   B. v = tan⁻¹(y/x), f = 1/z
   C. v = log r, f = z
   D. v = xy, f = z²
   Answer: A. f = ln r + iθ = log z. (Key point: Result)

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