Calculating Residue of cos(z)/z in Contour Integral

  • Thread starter Thread starter HappyEuler2
  • Start date Start date
  • Tags Tags
    Residue
Click For Summary
SUMMARY

The discussion focuses on calculating the contour integral \(\oint \frac{\cos(z)}{z}\) around the circle defined by \(|z|=1.5\). The integral utilizes the Cauchy Integral Formula and the Residue Theorem, with the residue at the pole \(z=0\) being determined. The residue is confirmed to be 1, leading to the final result of the integral being \(2\pi i\). The user sought clarification on the residue calculation process, which involves evaluating \(\lim_{z \to 0} (z-0) \frac{\cos(z)}{z}\).

PREREQUISITES
  • Understanding of contour integrals and complex analysis
  • Familiarity with the Cauchy Integral Formula
  • Knowledge of the Residue Theorem
  • Basic proficiency in using mathematical software like Maple
NEXT STEPS
  • Study the Cauchy Integral Formula in detail
  • Learn how to apply the Residue Theorem to various functions
  • Explore the use of Maple for complex analysis calculations
  • Review examples of calculating residues for different types of poles
USEFUL FOR

Students and professionals in mathematics, particularly those studying complex analysis, as well as anyone interested in mastering contour integration techniques.

HappyEuler2
Messages
4
Reaction score
0

Homework Statement


So the problem at hand is to calculate the contour integral \oint cos(z)/z around the circle abs(z)=1.5 .


Homework Equations


The integral is going to follow from the Cauchy-Integral Formula and the Residue theorem. The problem I am having is figuring out what the residue is going to be.


The Attempt at a Solution



So I know the pole is at z = 0, which lies inside of the contour. So the integral reduces to I=2*pi*i*residue @ 0. What I can't figure out is how to determine the residue. If I use maple, I know that the residue is 1, but I want to figure out where it comes from it. Any help?
 
Physics news on Phys.org
If f(z) has a simple pole at z=c then the residue is lim z->c of (z-c)*f(z), isn't it? What does that give you?
 
Ah, thank you. I just needed someone to write it out clearly for me.

So the formal answer =2*Pi*i
 
Last edited:

Similar threads

Complex Integration Along Given Path
  • · Replies 3 ·
Replies
3
Views
2K
Can I Use Substitution to Prove the Residue Theorem Integral?
  • · Replies 5 ·
Replies
5
Views
2K
Find the residues of the following function + Cauchy Residue
  • · Replies 2 ·
Replies
2
Views
2K
Residue Theorem: Finding the Integral of z^3e^(-1/z^2) over |z|=5
  • · Replies 6 ·
Replies
6
Views
2K
Calculating Integral Using Residue Theorem & Complex Variables
  • · Replies 4 ·
Replies
4
Views
2K
Compute the residue of a function
  • · Replies 2 ·
Replies
2
Views
2K
Complex Integration using residue theorem
  • · Replies 1 ·
Replies
1
Views
2K
Contour integral using residue theorem
  • · Replies 4 ·
Replies
4
Views
3K
What is the residue of f(z) = e^(-2/z^2) using a laurent series?
  • · Replies 2 ·
Replies
2
Views
1K
Can the contour integral of z⁷ be simplified using a parameterized expression?
  • · Replies 1 ·
Replies
1
Views
2K