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

Residue Theorem.

## The Attempt at a Solution

I just need help figuring out the circle C I'll be using. Suggestions?

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vela
Staff Emeritus
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What does the presence of z2/3 tell you?

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

vela
Staff Emeritus
Homework Helper
Yes, other than that. In particular, what's the effect of the fractional power?

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

vela
Staff Emeritus
Homework Helper
Right. Do you know what a branch point and a branch cut are?

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?

vela
Staff Emeritus
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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.

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?

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

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.

vela
Staff Emeritus
Homework Helper
Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.

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.

vela
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.