---
title: "Evaluate ∮z/((z-1)(z-3)) on |z|=3/2"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:cif-z-over-z-1-z-3"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/cif-z-over-z-1-z-3
site: BCT Engineering
---

# Evaluate ∮z/((z-1)(z-3)) on |z|=3/2

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2069 Poush
- Video: [if C is a Circle of |z|=3/2 Evaluate  integral 4-3z/z(z-1)(z-2) dz #complex analysis](https://www.youtube.com/watch?v=Q9eoGoxZw6E) — Maan Shree 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/cif-z-over-z-1-z-3
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Using Cauchy integral formula to evaluate: ∮_C z dz/((z - 1)(z - 3)) where C: |z| = 3/2.

## Short answer

For C:|z|=3/2, only z=1 lies inside. Let f(z)=z/(z-3), analytic inside C. By Cauchy integral formula, ∮_ C z/(( z-1)(z-3)) dz = 2πi f(1) = 2πi·1/(1−3) = −πi.

## Answer

**Cauchy's integral formula:** f(a) = (1/2πi) ∮_C f(z)/(z − a) dz for *a* inside C, with f analytic inside and on C.

#### Solution
C: |z| = 3/2. The pole z = 1 lies inside (1 < 3/2) and the pole z = 3 lies outside (3 > 3/2).

Write the integrand as

z/((z − 1)(z − 3)) = [ z/(z − 3) ] · 1/(z − 1).

Let f(z) = z/(z − 3). It is analytic inside and on C, since its only singularity z = 3 is outside C.

By CIF with a = 1:

∮_C z dz/((z − 1)(z − 3)) = 2πi f(1) = 2πi · 1/(1 − 3) = 2πi · (−1/2) = −πi

**Answer: −πi**

## Check yourself

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

1. For |z|=3/2, inside is:
   A. both
   B. none
   C. z = 3
   D. z = 1
   Answer: D. 3 > 3/2. (Key point: Inside)

2. Take f(z) =
   A. z/(z−3)
   B. z
   C. z/(z−1)
   D. 1/(z−3)
   Answer: A. Analytic inside. (Key point: f)

3. f(1) =
   A. −1/2
   B. −1
   C. 1/2
   D. 1
   Answer: A. 1/(1−3). (Key point: f(1))

4. Integral =
   A. 2πi
   B. πi
   C. −2πi
   D. −πi
   Answer: D. 2πi(−1/2). (Key point: Result)

## Close variants set in other exams

- [Evaluate ∮e^{2z}/((z-1)(z-2)) by CIF](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acif-e2z-z-1-z-2) — Applied Mathematics, chapter 1, asked 8×
- [Evaluate ∮(4-3z)/(z(z-1)(z-2)) on |z|=3/2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acif-4-3z-over-z-z-1-z-2) — Applied Mathematics, chapter 1, asked 6×

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