Problem with contour integration

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Homework Help Overview

The discussion revolves around evaluating the integral \(\int \frac{dx}{x^2 + 2x + 2}\) using contour integration techniques. Participants are exploring the identification of poles and residues related to the function \(f(z) = \frac{1}{z^2 + 2z + 2}\).

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the process of finding poles by solving the quadratic equation \(z^2 + 2z + 2 = 0\) and express confusion about the subsequent steps involving residues. There are questions about the correct contour to use and the evaluation of residues at the identified poles.

Discussion Status

Some participants have provided guidance on finding the roots of the quadratic equation and the need to evaluate residues. However, there is a lack of consensus on the correct approach to obtain the expected result of \(\pi\), with participants seeking clarification on their calculations and methods.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the amount of direct assistance they can receive. There is an emphasis on understanding the steps involved in contour integration and residue calculation without providing complete solutions.

Toyona10
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Homework Statement


Evaluate \intdx/(x2+2x+2) (limits from -∞ to ∞) by contour integration

Ok so, this is what i did...:

we have ---> \Gamma : |z|=R

\ointdz/(z2+2z+2) =
\intdx/x2+2x+2 (limits -R to R) +\intdz/(z2+2z+2) (lower limit \Gamma)--> (1)

[where f(z)=1/(z2+2z+2)]

we have, limz→∞ zf(z) = limz→∞ z/(z2+2z+2)
= 0
therefore,
limR→∞ \intdz/(z2+2z+2) [lower limit\Gamma
= 0

Taking the limit R→∞ on both sides of (1)

\ointdz/(z2+2z+2)= \intdx/x2+2x+2 [limits from -∞ to ∞) + 0


where f(z)= 1/z2+2z+2 so for finding out the poles of f(z)...
yeah this is where I am stuck, because in the previous one we did f(z) was→1/z4+1

so there we solved for z the way we solve for the roots of a complex number...
all that k= n-1 and nth root of z = rei(θ+2k∏)1/n...

so how do we solve in this case?

Thanks a lot in advance...
 
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To find the poles of 1/(z^2+2z+2) you need to find the roots of z^2+2z+2=0. That shouldn't be hard for you. It's a quadratic equation.
 
Dick said:
To find the poles of 1/(z^2+2z+2) you need to find the roots of z^2+2z+2=0. That shouldn't be hard for you. It's a quadratic equation.

okay...after solving quadratically, then we go for the residues, right? But i don't get the answer I am supposed to...its supposed to be π...
 
Toyona10 said:
okay...after solving quadratically, then we go for the residues, right? But i don't get the answer I am supposed to...its supposed to be π...

Tell us what you did!
 
Toyona10 said:
okay...after solving quadratically, then we go for the residues, right? But i don't get the answer I am supposed to...its supposed to be π...

Sure, it's pi. Show why you didn't get pi. What are the residues and what kind of contour are you picking? Can you show your work?
 
micromass said:
Tell us what you did!

I completed the square for that qudratic equation and its like this...→(z+1)2+1
then what? we find the residue for z→-1 to the order of 2?...
 
Toyona10 said:
I completed the square for that qudratic equation and its like this...→(z+1)2+1
then what? we find the residue for z→-1 to the order of 2?...

You find the roots where the poles are and pick a contour first. Then decide what residues you need to evaluate.
 

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