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∫dx/((x^(2/3)(x+1)), integrated over [0,∞] 
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#1
Feb512, 09:10 PM

P: 909

1. The problem statement, all variables and given/known data
As in thread title. 2. Relevant equations Residue Theorem. 3. The attempt at a solution I just need help figuring out the circle C I'll be using. Suggestions? 


#2
Feb612, 04:06 AM

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What does the presence of z^{2/3} tell you?



#4
Feb612, 10:03 AM

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∫dx/((x^(2/3)(x+1)), integrated over [0,∞]
Yes, other than that. In particular, what's the effect of the fractional power?



#5
Feb612, 10:27 AM

P: 909

Change the angle between z and the xaxis from ø to 2ø/3 


#6
Feb612, 10:53 AM

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Right. Do you know what a branch point and a branch cut are?



#8
Feb612, 12:03 PM

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It's more that you want to avoid crossing the branch cut than avoiding z=0, and you obviously want a piece or pieces of the contour to correspond to the original integral.



#9
Feb612, 12:22 PM

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#10
Feb612, 12:36 PM

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No, that's too complicated. Take a look at http://en.wikipedia.org/wiki/Methods...93_branch_cuts.
Also, rewrite the integrand as $$\frac{z^{1/3}}{z(z+1)}$$to make it clear how to calculate the residue at z=0. 


#12
Feb612, 02:16 PM

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Doesn't the answer to that question depend on which way Pacman is moving?



#13
Feb612, 05:21 PM

P: 909

But yeah, how am I gonna do this? I need C to be formed from a series of paths, each of which will have a line integral that approaches a real value after I take some limit. 


#14
Feb612, 06:04 PM

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Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.



#15
Feb612, 06:24 PM

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#16
Feb812, 02:47 PM

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That's what you're supposed to figure out. Did you understand the example on Wikipedia? That's pretty much the recipe you want to follow.



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