---
title: "Analytic function from u = 3x²y - y³"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:analytic-function-3x2y-y3"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/analytic-function-3x2y-y3
site: BCT Engineering
---

# Analytic function from u = 3x²y - y³

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2071 Magh, 5 marks
- Video: [How to Find Analytic Function When Harmonic Function is Given](https://www.youtube.com/watch?v=nlO8SAR9m3U) — Ekeeda
- 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-3x2y-y3
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Determine the analytic function f(z) = u + iv if u = 3x²y - y³.

## Short answer

Since u=3x²y−y³ is harmonic, the analytic function is f(z)=−i z³+c with v=3xy²−x³+constant.

## Answer

u = 3x²y − y³.

#### Check that u is harmonic
```text
u_x  = 6xy          u_xx = 6y
u_y  = 3x² − 3y²    u_yy = −6y
```

u_xx + u_yy = 0, so u is harmonic and an analytic f with Re f = u exists.

#### Milne-Thomson method
f′(z) = u_x − i u_y. Put x = z, y = 0:

f′(z) = 0 − i(3z² − 0) = −3i z²

Integrate: f(z) = −i z³ + c.

#### Find v
z³ = (x³ − 3xy²) + i(3x²y − y³), so

−i z³ = (3x²y − y³) − i(x³ − 3xy²)

Real part = 3x²y − y³ = u ✓, and v = −(x³ − 3xy²) = 3xy² − x³ + constant.

Check with C–R: v_y = 6xy = u_x ✓ and v_x = 3y² − 3x² = −u_y ✓.

**Answer: f(z) = −i z³ + c, with v = 3xy² − x³ + constant.**

## Check yourself

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

1. u_xx + u_yy is:
   A. 0
   B. 12y
   C. −12y
   D. 6
   Answer: A. 6y − 6y. (Key point: Harmonic)

2. f′(z) = u_x − i u_y at (z, 0) equals:
   A. 3i z²
   B. 3z²
   C. −3z²
   D. −3i z²
   Answer: D. u_x = 0, u_y = 3z². (Key point: f′)

3. f(z) =
   A. i z³ + c
   B. −i z³ + c
   C. z³ + c
   D. −z³ + c
   Answer: B. Integrate. (Key point: Result f)

4. The imaginary part v is:
   A. x³ + 3xy²
   B. 3xy² − x³ + const
   C. 3x²y − y³
   D. x³ − 3xy² + const
   Answer: B. −i z³ has imaginary part −(x³ − 3xy²). (Key point: v)

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