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

  • Thread starter Jamin2112
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  • #1
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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?
 

Answers and Replies

  • #2
vela
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What does the presence of z2/3 tell you?
 
  • #3
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What does the presence of z2/3 tell you?
Other than that there's a pole at z=0?
 
  • #4
vela
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Yes, other than that. In particular, what's the effect of the fractional power?
 
  • #5
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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
 
  • #6
vela
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Right. Do you know what a branch point and a branch cut are?
 
  • #7
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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?
 
  • #8
vela
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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
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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.
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?
 
  • #12
vela
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Doesn't the answer to that question depend on which way Pacman is moving?
 
  • #13
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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.
 
  • #14
vela
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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
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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.
 
  • #16
vela
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That's what you're supposed to figure out. :smile: Did you understand the example on Wikipedia? That's pretty much the recipe you want to follow.
 

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