Complex Integration Function with multiple poles at origin

[VVS]

Hello,

I hope somebody can help me with this one.

Homework Statement

I want to find the integral of 1/x^N*exp(ix) from -inf to inf.

Homework Equations

It is very likely that this can somehow be solved by using Cauchy's integral formula.

The Attempt at a Solution

I tried to integrate it by defining a countour as follows:
1. From -R to -r
2. semicircle from -r to +r around the origin from below
3. from r to R
4. Semicircle from +R to -R around the origin from above.

I can show that 4 tends to 0 as R tends to infinity
But I can't somehow evaluate 2:
I get 1/(r*exp(i*theta)^N*exp(i*r*exp(i*theta))r*exp(i*theta)d(theta)
Nothing cancels as nicely as in the case of a simple pole.

Thank you

 

[VVS]

sorry i didnt solve it

 

[Ray Vickson]

Hello,

I hope somebody can help me with this one.

Homework Statement

I want to find the integral of 1/x^N*exp(ix) from -inf to inf.

Homework Equations

It is very likely that this can somehow be solved by using Cauchy's integral formula.

The Attempt at a Solution

I tried to integrate it by defining a countour as follows:
1. From -R to -r
2. semicircle from -r to +r around the origin from below
3. from r to R
4. Semicircle from +R to -R around the origin from above.

I can show that 4 tends to 0 as R tends to infinity
But I can't somehow evaluate 2:
I get 1/(r*exp(i*theta)^N*exp(i*r*exp(i*theta))r*exp(i*theta)d(theta)
Nothing cancels as nicely as in the case of a simple pole.

Thank you

The integral may not converge. Since you have an improper integral, it needs to be *defined*, typically in terms of some limiting operations such as
$$\int_{-\infty}^{\infty} \frac{e^{ix}}{x^N} \, dx = \lim_{L,U \to \infty,\: a,b \to 0+} \left[ \int_{-L}^{-a} \frac{e^{ix}}{x^N} \, dx + \int_{b}^{U} \frac{e^{ix}}{x^N} \, dx \right].$$ Does this limit exist in your case?

 

[VVS]

Hey,
Thanks for your help but I had solved the problem now before your answer.
I called In=1/x^n*exp(ix). And I used integration by parts. I chose 1/x^n as the function to be integrated and exp(ix) as the function to be differentiated. That means that I get something proportiional to 1/x^(n-1)exp(ix) which is In-1. So I keep expressing In in terms of In-1, In-2 and so forth till I get to 1/x*exp(ix) which is easily integrated to Pi*i. All other terms vanish because they are of the from 1/x^N-n and the limits are +inf and-inf.
Thank you again for your help

 

[Ray Vickson]

Hey,
Thanks for your help but I had solved the problem now before your answer.
I called In=1/x^n*exp(ix). And I used integration by parts. I chose 1/x^n as the function to be integrated and exp(ix) as the function to be differentiated. That means that I get something proportiional to 1/x^(n-1)exp(ix) which is In-1. So I keep expressing In in terms of In-1, In-2 and so forth till I get to 1/x*exp(ix) which is easily integrated to Pi*i. All other terms vanish because they are of the from 1/x^N-n and the limits are +inf and-inf.
Thank you again for your help

I tried to warn you but you refused to listen.

 

[VVS]

Hey, The integrals converge with those limits. So I did listen to you.

 

[Ray Vickson]

Hey, The integrals converge with those limits. So I did listen to you.

All you did was to perform a sequence of illegal operations to obtain a wrong answer. I will say it only one more time: you need to look at limits when dealing with improper integrals.