---
title: "Conformal mapping; bilinear map to -3,-1,1"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:conformal-bilinear-0-1-inf"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/conformal-bilinear-0-1-inf
site: BCT Engineering
---

# Conformal mapping; bilinear map to -3,-1,1

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2075 Baisakh, 1+4 marks
- Video: [Mapping Through Bilinear Transformation Problem 1 - Conformal Mapping - Engineering Mathematics 3](https://www.youtube.com/watch?v=rwIEKKPho7o) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/conformal-bilinear-0-1-inf
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define conformal mapping. Find the bilinear transformation which maps the points z = 0, 1, ∞ into w = -3, -1, 1 respectively.

## Short answer

A conformal mapping preserves angles between curves in magnitude and sense where f'(z) is nonzero. Using the three conditions, the bilinear transformation is w=(z-3)/(z+1), valid except at 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 bilinear transformation
Let w = (az + b)/(cz + d), ad − bc ≠ 0.

- z = ∞ → w = 1: a/c = 1, so a = c
- z = 0 → w = −3: b/d = −3, so b = −3d
- z = 1 → w = −1: a + b = −(c + d)

Put a = c and b = −3d: c − 3d = −c − d, so c = d. Hence a = c = d, and b = −3a.

Take a = 1: **w = (z − 3)/(z + 1).**

Check: z = 0 gives −3 ✓. z = 1 gives −2/2 = −1 ✓. z = ∞ gives 1 ✓. Here ad − bc = 4 ≠ 0, so the mapping is valid and conformal everywhere except at z = −1.

## Check yourself

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

1. A conformal mapping:
   A. Preserves areas
   B. Preserves lengths
   C. Maps lines to lines
   D. Preserves angles in magnitude and sense
   Answer: D. Where f′ ≠ 0. (Key point: Conformal)

2. z=∞ → w=1 gives:
   A. c = 0
   B. a = 0
   C. b = d
   D. a = c
   Answer: D. a/c = 1. (Key point: Relation)

3. From the three conditions, a = c = d and b =
   A. −a
   B. −3a
   C. a
   D. 3a
   Answer: B. b = −3d. (Key point: Solving)

4. The map is:
   A. w = (z − 3)/(z + 1)
   B. w = (z − 3)/(z − 1)
   C. w = (z + 1)/(z − 3)
   D. w = (z + 3)/(z + 1)
   Answer: A. Verified at the three points. (Key point: Result)

## Close variants set in other exams

- [Bilinear map of 1, i, -1 to i, -1, -i](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Abilinear-1-i-minus1-to-i-minus1-minus-i) — Applied Mathematics, chapter 1, asked 2×

## Related in Applied Mathematics

- [Conformal mapping; map 1, i, -1 to 0, 1, ∞](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aconformal-map-1-i-minus1) — Applied Mathematics, chapter 1
- [Bilinear map z = 0,1,∞ to -3,-1,1 with fixed points](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Abilinear-0-1-inf-fixed-points) — Applied Mathematics, chapter 1
- [Bilinear map of -i, 0, i to -1, i, 1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Abilinear-minus-i-0-i) — Applied Mathematics, chapter 1, asked 4×
- [Bilinear map preserves cross ratio](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Abilinear-invariant-cross-ratio) — Applied Mathematics, chapter 1
- [Linear fractional map 2i, -2, -2i to -2, -2i, 2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alinear-transformation-2i-minus2) — Applied Mathematics, chapter 1
- [Linear fractional map z = 0,1,-1 to i,2,4](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alinear-fractional-0-1-minus1) — Applied Mathematics, chapter 1

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