Problem with contour integration

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SUMMARY

The discussion focuses on evaluating the integral \(\int \frac{dx}{x^2 + 2x + 2}\) from \(-\infty\) to \(\infty\) using contour integration techniques. The user initially sets up the contour \(\Gamma: |z|=R\) and determines that the limit of the integral over the contour approaches zero as \(R\) approaches infinity. The main challenge arises in finding the poles of the function \(f(z) = \frac{1}{z^2 + 2z + 2}\), which requires solving the quadratic equation \(z^2 + 2z + 2 = 0\). The discussion emphasizes the importance of calculating residues to arrive at the expected result of \(\pi\).

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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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