## ∫dx/((x^(2/3)(x+1)), integrated over [0,∞]

1. The problem statement, all variables and given/known data

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?

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 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus What does the presence of z2/3 tell you?

 Quote by vela What does the presence of z2/3 tell you?
Other than that there's a pole at z=0?

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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?

 Quote by vela Yes, other than that. In particular, what's the effect of the fractional power?
Change the distance between z and the origin from r to r2/3
Change the angle between z and the x-axis from ø to 2ø/3

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Right. Do you know what a branch point and a branch cut are?

 Quote by vela Right. Do you know what a branch point and a branch cut are?
Yeah, I somehow need a loop that avoid z=-1 and z=0. Right?

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus 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.

 Quote by vela 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.
So I'd take R>1 and make a half circle of radius R in the upper half of the plane. Then I'd make two little half circles that jump over z=-1 and z=0. Then I'd look at ∫C f(z)dz as the sum of several integrals, one of which can written as a real-valued integral and see what happens as R→∞ and the radii of the little half circles go to zero. Right?

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus 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.

 Quote by vela 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.
So in my case the path should resemble a backwards Pacman?

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Doesn't the answer to that question depend on which way Pacman is moving?

 Quote by vela Doesn't the answer to that question depend on which way Pacman is moving?
I forgot that PacMan is in perpetual motion.

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.

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.

 Quote by vela Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.
How would that work? I want ∫f(x)dx (integrated on [0, R]) to be one of the four line integrals.

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus 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.