---
title: "Linear fractional map z = 0,1,-1 to i,2,4"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:linear-fractional-0-1-minus1"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/linear-fractional-0-1-minus1
site: BCT Engineering
---

# Linear fractional map z = 0,1,-1 to i,2,4

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2074 Bhadra, 1+4 marks
- 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-0-1-minus1
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define linear fractional mapping. Find the bilinear mapping which maps the points z = 0, 1, -1 to w = i, 2, 4.

## Short answer

A linear fractional mapping is w=(az+b)/(cz+d) with ad-bc≠0; it maps circles/lines to circles/lines and is determined by three points. Using the cross-ratio, the required mapping is w=[(8-3i)z+i]/[(3-i)z+1].

## Answer

#### Linear fractional mapping
A **linear fractional (bilinear, Möbius) mapping** is w = (az + b)/(cz + d), where a, b, c, d are complex constants and ad − bc ≠ 0. It maps circles and lines to circles or lines, and is fixed by three points and their images.

#### Solution
Take z₁ = 0, z₂ = 1, z₃ = −1 and w₁ = i, w₂ = 2, w₃ = 4.

Cross-ratio form of the bilinear transformation:

(w − w₁)(w₂ − w₃) / [(w − w₃)(w₂ − w₁)] = (z − z₁)(z₂ − z₃) / [(z − z₃)(z₂ − z₁)]

Right side: z(1 + 1)/[(z + 1)(1 − 0)] = 2z/(z + 1).

Left side: (w − i)(2 − 4)/[(w − 4)(2 − i)] = −2(w − i)/[(w − 4)(2 − i)].

```text
(w − i)/(w − 4) = −(2 − i) z/(z + 1)  =: S
w − i = S(w − 4)   →   w = (i − 4S)/(1 − S)
```

Substitute S = −(2 − i)z/(z + 1) and multiply numerator and denominator by (z + 1):

```text
numerator   = i(z + 1) + 4(2 − i)z = (8 − 3i) z + i
denominator = (z + 1) + (2 − i) z  = (3 − i) z + 1
```

**Answer: w = [(8 − 3i) z + i] / [(3 − i) z + 1]**

#### Check
- z = 0: w = i/1 = i ✓
- z = 1: (8 − 3i + i)/(3 − i + 1) = (8 − 2i)/(4 − i) = 2 ✓
- z = −1: (−8 + 3i + i)/(−3 + i + 1) = (−8 + 4i)/(−2 + i) = 4 ✓

## Check yourself

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

1. Bilinear map is determined by:
   A. Three points and their images
   B. Its derivative
   C. Two points only
   D. One point
   Answer: A. Standard. (Key point: Definition)

2. Right side z(1+1)/[(z+1)(1−0)] equals:
   A. 2z/(z+1)
   B. z/2
   C. z/(z+1)
   D. 2z/(z−1)
   Answer: A. Substitute z1=0, z2=1, z3=−1. (Key point: Cross ratio RHS)

3. The answer maps z=0 to:
   A. 2
   B. 4
   C. 0
   D. i
   Answer: D. w = i/1. (Key point: Check z=0)

4. The answer is:
   A. [(8−3i)z + i]/[(3−i)z + 1]
   B. [(8−3i)z − i]/[(3−i)z − 1]
   C. [(8+3i)z + i]/[(3+i)z + 1]
   D. (2z + i)/(z + 1)
   Answer: A. Checked at the 3 points. (Key point: Result)

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