---
title: "Image of a circle under w = 1/z"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:image-of-a-circle-under-w"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/image-of-a-circle-under-w
site: BCT Engineering
---

# Image of a circle under w = 1/z

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2082 Kartik (new course), 2 marks; 2081 Chaitra (new course), 2 marks (asked 2 times)
- Video: [Inverse Mapping – Image of a Circle under w=1/z ​ | Complex Analysis](https://www.youtube.com/watch?v=6a6jxBfm2FA) — Manimaran J
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/image-of-a-circle-under-w
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

What will be the image of the circle |z − i| = 1 under the mapping w = f(z) = 1/z in the complex plane?

## Short answer

The circle |z−i|=1 passes through the origin, mapping under w=1/z to the line Im(w) = −1/2. From x²+y²=2y, v = −y/(x²+y²) = −1/2.

## Answer

The circle |z − i| = 1 passes through the origin, so under w = 1/z it maps to a **straight line** (a circle through the origin maps to a line).

Let z = x + iy. The circle is x² + (y − 1)² = 1, that is

```text
x² + y² = 2y        ... (1)
```

Now w = 1/z = z̄/|z|² = (x − iy)/(x² + y²). So

```text
u = x/(x² + y²),   v = −y/(x² + y²)
```

Using (1), x² + y² = 2y, so

```text
v = −y/(2y) = −1/2
```

**Answer:** the image is the straight line v = −1/2 (the line Im(w) = −1/2, parallel to the real axis). The point z = 0 on the circle goes to w = ∞, so the whole line is covered. Check: z = 2i gives w = −i/2, and z = 1 + i gives w = (1 − i)/2; both have v = −1/2.

## Check yourself

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

1. Why is the image of |z - i| = 1 under w = 1/z a straight line?
   A. The circle has radius 1
   B. The circle is centered at i
   C. The mapping is linear
   D. The circle passes through the origin
   Answer: D. A circle through the origin maps to a line under inversion. (Key point: Why a line)

2. The circle in Cartesian form is?
   A. x^2 + y^2 = 2y
   B. x^2 + y^2 = 1
   C. x^2 + y^2 = 2x
   D. x^2 + y^2 = -2y
   Answer: A. x^2 + (y-1)^2 = 1 expands to x^2 + y^2 = 2y. (Key point: Circle equation)

3. What is the image?
   A. v = 1/2
   B. v = -1/2
   C. u = -1/2
   D. u^2 + v^2 = 1
   Answer: B. v = -y/(x^2+y^2) = -y/(2y) = -1/2. (Key point: Image)

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- [Stream function and complex potential](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Astream-function-and-complex-potential) — Mathematics III, chapter 2

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