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

[Jamin2112]

Homework Statement

As in thread title.

Homework Equations

Residue Theorem.

The Attempt at a Solution

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

 

[vela]

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

 

[Jamin2112]

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

Other than that there's a pole at z=0?

 

[vela]

Yes, other than that. In particular, what's the effect of the fractional power?

 

[Jamin2112]

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 r^2/3
Change the angle between z and the x-axis from ø to 2ø/3

 

[vela]

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

 

[Jamin2112]

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]

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.

 

[Jamin2112]

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]

No, that's too complicated. Take a look at http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28IV.29_.E2.80.93_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.

 

[Jamin2112]

No, that's too complicated. Take a look at http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28IV.29_.E2.80.93_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?

 

[vela]

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

 

[Jamin2112]

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 going to 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]

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

 

[Jamin2112]

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]

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.