---
title: "Find f(z) from u - v = eˣ(cos y - sin y)"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:analytic-function-u-minus-v"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/analytic-function-u-minus-v
site: BCT Engineering
---

# Find f(z) from u - v = eˣ(cos y - sin y)

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2069 Poush
- Video: [If f(z) =u+iv is analytic and u-v =e^x (cosy-siny), find f(z) || complex analysis bsc maths](https://www.youtube.com/watch?v=Ow3ATjUT6uk) — Swati Theng Mathematics
- 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-u-minus-v
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

If f(z) = u + iv is any analytic function of complex variable z and u - v = eˣ(cos y - sin y), find f(z) in terms of z.

## Short answer

Using (1+i)f(z), U=u−v=eˣ(cos y−sin y) is harmonic; then f(z)=eᶴ+c.

## Answer

Given u − v = eˣ (cos y − sin y).

#### Method
Form the analytic function F(z) = (1 + i) f(z):

(1 + i)(u + iv) = (u − v) + i(u + v) = U + iV

where U = u − v = eˣ(cos y − sin y) and V = u + v. Since F is analytic, U is harmonic and we can find F by the Milne-Thomson method.

```text
U_x = eˣ (cos y − sin y)
U_y = eˣ (−sin y − cos y)
```

F′(z) = U_x − i U_y. Put x = z, y = 0:

F′(z) = e^z (1 − 0) − i e^z (−0 − 1) = e^z + i e^z = (1 + i) e^z

Integrate: F(z) = (1 + i) e^z + c₁.

#### Recover f(z)
f(z) = F(z)/(1 + i) = e^z + c, where c = c₁/(1 + i) is a constant.

#### Check
For f = e^z = eˣ cos y + i eˣ sin y: u = eˣ cos y, v = eˣ sin y, so u − v = eˣ(cos y − sin y) ✓.

(The constant must make u − v unchanged, so c = β(1 + i) with β real; the simplest choice is c = 0.)

**Answer: f(z) = e^z + c**

## Check yourself

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

1. Which combination turns u − v into the real part of an analytic function?
   A. (1+i)f(z)
   B. f(z)²
   C. (1−i)f(z)
   D. i f(z)
   Answer: A. (1+i)(u+iv) = (u−v) + i(u+v). (Key point: Trick)

2. F′(z) at x=z, y=0 equals:
   A. (1−i) e^z
   B. (1+i) e^z
   C. e^z
   D. i e^z
   Answer: B. e^z + i e^z. (Key point: F′)

3. f(z) is recovered by:
   A. Taking F²
   B. Differentiating F
   C. Dividing F by (1+i)
   D. Multiplying F by (1+i)
   Answer: C. f = F/(1+i). (Key point: Recover f)

4. f(z) =
   A. z e^z
   B. e^z(1+i)
   C. e^{2z}
   D. e^z + c
   Answer: D. Check: u − v = e^x(cos y − sin y). (Key point: Result)

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