---
title: "Analytic function from u = log√(x²+y²)"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:analytic-function-log-sqrt"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/analytic-function-log-sqrt
site: BCT Engineering
---

# Analytic function from u = log√(x²+y²)

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2073 Magh, 1+4 marks; 2071 Bhadra (asked 2 times)
- Video: [Construction of analytic function | 𝒖=𝒍𝒐𝒈 √(𝒙^𝟐+𝒚^𝟐 ) | Complex analysis  18mat41| Dr Prashant Patil](https://www.youtube.com/watch?v=bTI_EGevnHo) — Dr Prashant Patil
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/analytic-function-log-sqrt
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define analytic function. Determine the analytic function f(z) = u + iv if u = log√(x² + y²).

## Short answer

An analytic function is differentiable at a point and in its neighbourhood. For u = log√(x²+y²), using Milne-Thomson method: f′(z)=u_x−i u_y =1/z, so f(z)=log z + c, with harmonic conjugate v = tan⁻¹(y/x) + constant.

## Answer

#### Definition
A function f(z) is **analytic** at a point if it is differentiable at that point and in some neighbourhood of it. It is analytic in a region if it is analytic at every point of the region.

#### Solution
u = log√(x² + y²) = ½ log(x² + y²).

```text
u_x = x/(x² + y²)
u_y = y/(x² + y²)
```

(u is harmonic: u_xx + u_yy = (y² − x²)/(x² + y²)² + (x² − y²)/(x² + y²)² = 0.)

**Milne-Thomson method.** Since f′(z) = u_x − i u_y, and the result holds for all x, y, put x = z, y = 0:

f′(z) = u_x(z, 0) − i u_y(z, 0) = 1/z − i·0 = 1/z

Integrate: f(z) = log z + c, where c is a constant.

#### Find v
log z = ln r + iθ, so f(z) = ln√(x² + y²) + i tan⁻¹(y/x) + c.

```text
u = log√(x² + y²)
v = tan⁻¹(y/x) + constant
```

Check with the C–R equations: v_y = x/(x² + y²) = u_x, and v_x = −y/(x² + y²) = −u_y.

**Answer: f(z) = log z + c, with v = tan⁻¹(y/x) + c′.**

## Check yourself

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

1. u = log√(x²+y²) equals:
   A. ½ log(x²+y²)
   B. √(log(x²+y²))
   C. log(x²+y²)
   D. 2 log(x²+y²)
   Answer: A. Log of a square root halves the log. (Key point: Simplify)

2. In the Milne-Thomson method, f′(z) is found by:
   A. u_x + i u_y with x = z, y = 0
   B. u_y − i u_x with x = z, y = 0
   C. u_x − i u_y with x = z, y = 0
   D. u_x − i u_y with x = 0, y = z
   Answer: C. f′ = u_x − i u_y and we substitute x=z, y=0. (Key point: Milne-Thomson)

3. Here f′(z) equals:
   A. log z
   B. z
   C. 1/z²
   D. 1/z
   Answer: D. u_x(z,0) = z/z² = 1/z and u_y(z,0) = 0. (Key point: f′)

4. The analytic function is:
   A. 1/z + c
   B. z log z
   C. e^z
   D. log z + c
   Answer: D. Integrating 1/z gives log z; the imaginary part is tan⁻¹(y/x). (Key point: Result)

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