---
title: "Linear fractional map -2, 0, 2 to ∞, 1/2, 3/4"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:linear-fractional-minus2-0-2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/linear-fractional-minus2-0-2
site: BCT Engineering
---

# Linear fractional map -2, 0, 2 to ∞, 1/2, 3/4

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2069 Bhadra
- Video: [Bilinear Mapping or Transformation II Mobuis Transformation II Linear Fractional Transformation](https://www.youtube.com/watch?v=OCc-rQkpRKA) — Study Buddy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/linear-fractional-minus2-0-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Find linear fractional transformation mapping of: -2 ↦ ∞, 0 ↦ 1/2, 2 ↦ 3/4.

## Short answer

Let w=(az+b)/(cz+d). The conditions give d=2c, c=b, and a=1 with b=c=1, so w=(z+1)/(z+2).

## Answer

Find w = (az + b)/(cz + d) with −2 → ∞, 0 → 1/2, 2 → 3/4.

#### Conditions
- z = −2 maps to ∞: the denominator vanishes, −2c + d = 0, so d = 2c.
- z = 0 maps to 1/2: b/d = 1/2, so d = 2b. Hence 2c = 2b and c = b.
- z = 2 maps to 3/4: (2a + b)/(2c + d) = 3/4.

Take b = c = 1. Then d = 2. The third condition gives (2a + 1)/(2 + 2) = 3/4, so 2a + 1 = 3 and a = 1.

**Answer: w = (z + 1)/(z + 2)**

#### Check
- z = −2: denominator 0, so w = ∞ ✓
- z = 0: 1/2 ✓
- z = 2: 3/4 ✓

Also ad − bc = 2 − 1 = 1 ≠ 0, so the transformation is non-degenerate.

## Check yourself

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

1. −2 ↦ ∞ means:
   A. c = 0
   B. −2c + d = 0
   C. −2a + b = 0
   D. a = 0
   Answer: B. Denominator vanishes. (Key point: ∞ image)

2. 0 ↦ 1/2 means:
   A. a = b
   B. a/c = 1/2
   C. b/d = 1/2
   D. b = 2d
   Answer: C. w(0) = b/d. (Key point: 0 image)

3. From conditions, d = 2c and c =
   A. b
   B. a
   C. 0
   D. 2b
   Answer: A. d = 2b so c = b. (Key point: Relations)

4. w =
   A. (z+2)/(z+1)
   B. (z+1)/(z−2)
   C. (z+1)/(z+2)
   D. (z−1)/(z−2)
   Answer: C. z=2: 3/4. (Key point: Result)

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