Possible Values of Complex Integration for Contours in D and from 0 to 1

Click For Summary

Homework Help Overview

The discussion revolves around evaluating complex integrals involving the function \( \frac{1}{1+z^2} \) over different contours in the complex plane, specifically in the context of the domain \( D = \mathbb{C} \setminus \{-i, i\} \) and from 0 to 1.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to evaluate the integral around a closed contour and questions whether the absence of poles in the domain affects the evaluation. They also explore the implications of Cauchy's theorem on the integral's value.
  • Participants raise questions about the nature of the contour from 0 to 1, considering various shapes and orientations, and whether the choice of contour impacts the integral's evaluation.
  • One participant suggests considering different scenarios regarding the winding of the contour around the poles and the implications for the integral's value.
  • Another participant expresses uncertainty about the residue theorem, indicating it has not yet been covered in their studies.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the problem and the implications of their assumptions. Some guidance has been offered regarding the relationship between the contour's path and the poles, but there is no explicit consensus on the evaluation of the integrals.

Contextual Notes

Participants note that the residue theorem has not yet been covered in their coursework, which may limit their ability to fully engage with the problem's complexities. There is also uncertainty regarding the representation of contours and their effects on the integral's evaluation.

oddiseas
Messages
66
Reaction score
0

Homework Statement



1)a Let D = C\{-i,i} and let γ be a closed contour in D. Find all the possible
values of :

(∫(1/(1+z²))dz around γ)

b)
If σ is a contour from 0 to 1, determine all possible
values of:
(∫(1/(1+z²))dz ( around σ)

Homework Equations





The Attempt at a Solution


for part a the denominator factorises to (z+i)(z-i). Now these points are NOT in the domain. If they are not in the domain does that mean i can't evaluate the integral or does it mean that i can evaluate the integral because the points i and -i are discontinuities, but since f(z) is analytic in the domain i can evaluate it and it should give me zero, by cauchys theorem. Anyway when i evaluate it i get the following:

((-1)/(2i)){∫(1/(z+i))-∫(1/(z-i)) and this gives me zero but i am not sure if i am right.


b)
If σ is a contour from 0 to 1, determine all possible
values of:
(∫(1/(1+z²))dz ( around σ)

I am not sure how to try this, or how to represent a contour from zero to 1. Should the contour be a quarter circle moving clockwise, or should it be a line on the real axis? or doesn't it matter what contour i use as long as the endpoints agree. When i calcvulated this using the partial fraction representation above i get zero again. So i think i am stuffing up somewhere. Can someone show me the procedure they use forevaluating these integrals
 
Physics news on Phys.org
Suppose your contour winds around one of the poles +/-i in a counterclockwise direction? Suppose it winds around twice. Suppose it winds around one clockwise? Suppose you don't wind around any poles? Etc etc. There are lots of possible values for both questions. Think about the Cauchy integral formula and the residue theorem.
 
we havnt done the residue theorem. we are doing it at the end of next week;
I am thinking that if part a is equal to 0, then it can take on any 2pi multiple of this. Am i on the right trach?
 
Roughly. Can you say why you would think that exactly, using the Cauchy integral formula, maybe?
 

Similar threads

Complex Integration Along Given Path
  • · Replies 3 ·
Replies
3
Views
2K
Can the contour integral of z⁷ be simplified using a parameterized expression?
  • · Replies 1 ·
Replies
1
Views
2K
Can I Use Substitution to Prove the Residue Theorem Integral?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Evaluate two complex integrals along a parabolic contour
  • · Replies 2 ·
Replies
2
Views
1K
Contour Integrals: Working Check
  • · Replies 8 ·
Replies
8
Views
2K
How Do You Solve a Complex Integral Using Cauchy-Goursat's Theorem?
  • · Replies 32 ·
2
Replies
32
Views
4K
Contour Integration over Square, Complex Anaylsis
  • · Replies 3 ·
Replies
3
Views
3K
Complex integration is giving the wrong answer by a factor of two
  • · Replies 13 ·
Replies
13
Views
2K
Contour integration around a square
  • · Replies 26 ·
Replies
26
Views
5K
Contour integral- Complex variables
  • · Replies 4 ·
Replies
4
Views
2K