Complex Analysis Homework: Calculating Integral

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Homework Help Overview

The discussion revolves around calculating a complex integral using the residue theorem in the context of complex analysis. The original poster describes their attempts at substitution and finding residues at specific points.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts substitution and residue calculation but expresses uncertainty about their approach, particularly regarding the treatment of singular points. Some participants question the validity of the substitution used and the choice of contour for integration.

Discussion Status

Participants are actively engaging with the original poster's approach, raising questions about the substitution and the contour path for integration. There is no explicit consensus yet, but the discussion is exploring various interpretations and methods.

Contextual Notes

The original poster's approach involves a removable singularity and considerations about the choice of contour for the integral, which may affect the outcome. There is an emphasis on ensuring the correct treatment of roots in the substitution process.

asi123
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Homework Statement



Hey guys.

So, I need to calculate this integral. I uploaded what I tried to do.
First of all, did the substitute, then I tried to use the residue theorem so I was looking for the residue of z=0 which is happen to be a removable singular point so it's just 0, then I went for the z=2*pi*k (when k can't be 0) residue and found out that it's 0.
I guess I have a mistake there, any idea guys?

Thanks.


Homework Equations





The Attempt at a Solution

 

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Your substitutions a little fishy; does [tex]e^{z}=x^2+1[/tex] really mean that [tex]x=\sqrt{e^z-1}[/tex]? How are you excluding the negative root?
 
If you are going to use residues, what close path are you going to integrate over?
 
HallsofIvy said:
If you are going to use residues, what close path are you going to integrate over?

I thought about a closed contour consisting of the semi-circle with radius r and centre at z = 0 and the line segment going from z = -r to z = r and then doing the r --> oo thing. That way, I'll only have one pole, p=i.
Then, I'll try to break it into -r to 0 and from 0 to r, and find the latter, does that seems right?

Thanks.
 

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