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

# Analytic function from u = y³ - 3x²y

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2073 Bhadra, 1+4 marks
- Video: [Complex Analysis - Short Trick To Find Harmonic Conjugate By GP Sir](https://www.youtube.com/watch?v=OQQqbV32b78) — Dr.Gajendra Purohit
- 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-y3-3x2y
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define harmonic function of complex variable. Determine the analytic function f(z) = u + iv if u = y³ - 3x²y.

## Short answer

A harmonic function satisfies Laplace's equation u_xx+u_yy=0. For u=y³−3x²y, the analytic function is f(z)=i z³+c with v=x³−3xy²+constant.

## Answer

#### Harmonic function
A real function u(x, y) with continuous second partial derivatives is **harmonic** if it satisfies Laplace's equation: u_xx + u_yy = 0. The real and imaginary parts of an analytic function are harmonic.

#### Solution
u = y³ − 3x²y.

```text
u_x  = −6xy         u_xx = −6y
u_y  = 3y² − 3x²    u_yy = 6y
```

u_xx + u_yy = 0, so u is harmonic and a conjugate v exists.

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

f′(z) = 0 − i(0 − 3z²) = 3i z²

Integrating: f(z) = i z³ + c.

#### Find v
Expand z³ = (x + iy)³ = (x³ − 3xy²) + i(3x²y − y³).

i z³ = −(3x²y − y³) + i(x³ − 3xy²) = (y³ − 3x²y) + i(x³ − 3xy²)

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

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

## Check yourself

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

1. u_xx + u_yy for u = y³ − 3x²y is:
   A. 6
   B. 0
   C. −12y
   D. 12y
   Answer: B. −6y + 6y = 0. (Key point: Harmonic)

2. u_x and u_y are:
   A. −6xy and 3y² − 3x²
   B. −6xy and 3x² − 3y²
   C. 3y² and −6x
   D. 6xy and 3y² − 3x²
   Answer: A. Direct differentiation. (Key point: Derivatives)

3. f′(z) = u_x − i u_y at x=z, y=0 is:
   A. −3i z²
   B. 3i z²
   C. 6z
   D. 3z²
   Answer: B. 0 − i(−3z²) = 3iz². (Key point: f′)

4. The analytic function is:
   A. z³ + c
   B. −i z³ + c
   C. i z³ + c
   D. i z² + c
   Answer: C. ∫3iz² dz = iz³. (Key point: Result)

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