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

# Evaluate ∮e^z/((z-1)(z-3)) on |z|=2

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2068 Bhadra
- Video: [#15 || Problem#3||Cauchy’s theorem ||∫𝒅𝒛/(𝒛^𝟐−𝟒) || 𝒄:|𝒛|=𝟏 || 𝒄:|𝒛|=𝟑 || 𝒄: |𝒛+𝟐|=𝟏 || 18MAT41](https://www.youtube.com/watch?v=i8wkriYGMfM) — Math E Connect
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/cif-e-z-z-1-z-3
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Write the statement of Cauchy's integral formula. Use it to evaluate the integral ∮_C e^z/((z - 1)(z - 3)) dz where C is the circle |z| = 2.

## Short answer

Cauchy's integral formula: ∮_ C f(z)/(z-a) dz = 2πi f(a) if f is analytic inside and on simple closed C and a is inside. For ∮_ C e^z/(( z-1)(z-3)) dz with C: |z|=2, only pole z=1 is inside. Let f(z)=e^z/(z-3) analytic. Then integral = 2πi f(1) = 2πi(e/(−2)) = −πi e.

## Answer

#### Statement of Cauchy's integral formula
If f(z) is analytic inside and on a simple closed contour C and *a* is any point inside C, then

f(a) = (1/2πi) ∮_C f(z)/(z − a) dz,  that is, ∮_C f(z)/(z − a) dz = 2πi f(a).

#### Solution
C: |z| = 2. The poles are z = 1 (|1| < 2, inside) and z = 3 (|3| > 2, outside).

Write the integrand as e^z/(z − 3) · 1/(z − 1). Let f(z) = e^z/(z − 3). It is analytic inside and on C, because its only singularity is z = 3, which is outside C.

By CIF with a = 1:

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

**Answer: −πie ≈ −8.54 i**

## Check yourself

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

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

2. The choice of f is:
   A. e^z/(z−3)
   B. e^z/(z−1)
   C. 1/(z−3)
   D. e^z
   Answer: A. Analytic inside C. (Key point: f)

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

4. The integral is:
   A. πi e
   B. 2πi e
   C. −πi e
   D. −2πi e
   Answer: C. 2πi(−e/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×

## Related in Applied Mathematics

- [Cauchy theorem vs formula; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acif-distinguish-e-z-z-1-z-2) — Applied Mathematics, chapter 1, asked 4×
- [State and prove Cauchy's integral formula](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acauchy-integral-formula-prove) — Applied Mathematics, chapter 1, asked 5×
- [Define complex integral; evaluate ∮log z dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-log-z) — Applied Mathematics, chapter 1
- [Evaluate ∮(z+1)dz over a square](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-square-contour) — Applied Mathematics, chapter 1
- [State and prove Cauchy integral theorem](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acauchy-integral-theorem-prove) — Applied Mathematics, chapter 1, asked 3×
- [Evaluate ∫(x²+iy)dz along y = x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-path-y-x2) — Applied Mathematics, chapter 1

## Related in other subjects

- [Cauchy's formula and equilibrium equations](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Acauchy-formula-equilibrium) — Mechanics of Solids, chapter 2
- [Evaluate a line integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-a-line-integral-two) — Mathematics III, chapter 2
- [Evaluate a line integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-a-line-integral-three) — Mathematics III, chapter 2
- [Evaluate a line integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-a-line-integral-four) — Mathematics III, chapter 2