---
title: "Conformal mapping; map 1, i, -1 to 0, 1, ∞"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:conformal-map-1-i-minus1"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/conformal-map-1-i-minus1
site: BCT Engineering
---

# Conformal mapping; map 1, i, -1 to 0, 1, ∞

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2073 Bhadra, 1+4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/conformal-map-1-i-minus1
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

What do you mean by conformal mapping? Find the linear transformation which maps points z₁ = 1, z₂ = i, z₃ = -1 into the points w₁ = 0, w₂ = 1, w₃ = ∞.

## Short answer

A conformal mapping preserves angles between curves in magnitude and sense where f'(z)≠0. Since z=1 maps to 0 and z=-1 to infinity, use w=k(z-1)/(z+1); imposing z=i gives k=-i, so w=-i(z-1)/(z+1).

## Answer

A **conformal mapping** w = f(z) is a mapping that preserves angles between any two curves, both in magnitude and in sense (direction), at every point where f′(z) ≠ 0. Every analytic function with f′(z) ≠ 0 is conformal.

#### Finding the transformation
z₁ = 1, z₂ = i, z₃ = −1 map to w₁ = 0, w₂ = 1, w₃ = ∞.

Since w₁ = 0 corresponds to z = 1 and w₃ = ∞ corresponds to z = −1, the transformation has the form

w = k (z − 1)/(z + 1).

Use z = i, w = 1:

1 = k (i − 1)/(i + 1). Since (i − 1)/(i + 1) = [(i − 1)(1 − i)]/[(1 + i)(1 − i)] = 2i/2 = i, we get 1 = k·i, so k = 1/i = −i.

**Answer: w = −i (z − 1)/(z + 1) = i (1 − z)/(1 + z)**

(Cross-ratio check: with w₃ → ∞ the formula reduces to (w − w₁)/(w₂ − w₁) = (z − z₁)(z₂ − z₃)/[(z − z₃)(z₂ − z₁)], giving w = (z − 1)(i + 1)/[(z + 1)(i − 1)] = −i(z − 1)/(z + 1) since (i + 1)/(i − 1) = −i.)

Check: z = 1 gives 0 ✓. z = i gives −i·i = 1 ✓. z = −1 gives ∞ ✓.

## Check yourself

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

1. Since 1→0 and −1→∞, the map has the form:
   A. w = k(z+1)/(z−1)
   B. w = k(z−1)/(z+1)
   C. w = k/(z−1)
   D. w = k z
   Answer: B. Zero at z=1, pole at z=−1. (Key point: Form)

2. Using z=i, w=1 gives k =
   A. −i
   B. 1
   C. i
   D. −1
   Answer: A. (i−1)/(i+1) = i, so k = 1/i. (Key point: k)

3. (i−1)/(i+1) equals:
   A. −1
   B. 1
   C. i
   D. −i
   Answer: C. Multiply by conjugate. (Key point: Value)

4. w =
   A. −i(z−1)/(z+1)
   B. i(z−1)/(z+1)
   C. −i(z+1)/(z−1)
   D. (z−1)/(z+1)
   Answer: A. Check z=i: −i·i = 1. (Key point: Result)

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