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


by Jamin2112
Tags: , ∫dx or x2 or 3x, integrated
Jamin2112
Jamin2112 is offline
#1
Feb5-12, 09:10 PM
Jamin2112's Avatar
P: 864
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?
Phys.Org News Partner Science news on Phys.org
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
vela
vela is online now
#2
Feb6-12, 04:06 AM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
What does the presence of z2/3 tell you?
Jamin2112
Jamin2112 is offline
#3
Feb6-12, 09:14 AM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
What does the presence of z2/3 tell you?
Other than that there's a pole at z=0?

vela
vela is online now
#4
Feb6-12, 10:03 AM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534

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


Yes, other than that. In particular, what's the effect of the fractional power?
Jamin2112
Jamin2112 is offline
#5
Feb6-12, 10:27 AM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
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 to 2/3
vela
vela is online now
#6
Feb6-12, 10:53 AM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
Right. Do you know what a branch point and a branch cut are?
Jamin2112
Jamin2112 is offline
#7
Feb6-12, 12:00 PM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
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
vela is online now
#8
Feb6-12, 12:03 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
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
Jamin2112 is offline
#9
Feb6-12, 12:22 PM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
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
vela is online now
#10
Feb6-12, 12:36 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
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.
Jamin2112
Jamin2112 is offline
#11
Feb6-12, 12:42 PM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
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?
vela
vela is online now
#12
Feb6-12, 02:16 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
Doesn't the answer to that question depend on which way Pacman is moving?
Jamin2112
Jamin2112 is offline
#13
Feb6-12, 05:21 PM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
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
vela is online now
#14
Feb6-12, 06:04 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.
Jamin2112
Jamin2112 is offline
#15
Feb6-12, 06:24 PM
Jamin2112's Avatar
P: 864
Quote Quote by vela View Post
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
vela is online now
#16
Feb8-12, 02:47 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,534
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.


Register to reply

Related Discussions
Estimate ∫γ dz/(1 + z^4) as R→∞. Calculus & Beyond Homework 1
∫∫ x^2 dA ; bounded by ellipse Calculus & Beyond Homework 6
[math analysis] sup f< sup g==>∫f^n<∫g^n Calculus & Beyond Homework 3
E(X) < ∞ ? What does it mean? Set Theory, Logic, Probability, Statistics 19
[SOLVED] Does 0/0 and ∞/∞ = 1 ? Precalculus Mathematics Homework 23