How to do this integral ?

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In summary, you are trying to integrate an equation in which the imaginary part contains a singularity at x=0. You are also trying to change the variable to y, but this does not help. You need to find a way to rationalize the denominator or convert it to a complex exponential.
  • #1
hiyok
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hi,

I have difficulty in figuring out the following integral:

[itex] I(l,m;z) = \int^1_0 dx~\frac{x^l}{z - x^m} [/itex],

where [itex] l [/itex] and [itex] m [/itex] are integers, while [itex] z = \omega + i0_+[/itex] is a complex number that is infinitely close to the real axis. What is interesting to me is when [itex] \omega [/itex] is close to zero, so that the integrand bears a singularity in the domain.

Could somebody help me out ?

Thanks a lot !

hiyok
 
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  • #2
hiyok said:
hi,

I have difficulty in figuring out the following integral:

[itex] I(l,m;z) = \int^1_0 dx~\frac{x^l}{z - x^m} [/itex],

where [itex] l [/itex] and [itex] m [/itex] are integers, while [itex] z = \omega + i0_+[/itex] is a complex number that is infinitely close to the real axis.
You mean:$$z=\lim_{k\rightarrow 0^+}\omega + ik$$... but since z is not a function of x, what is the problem?

I'm more concerned with what ##x^\prime## represents.

What is interesting to me is when [itex] \omega [/itex] is close to zero, so that the integrand bears a singularity in the domain.
The point x=0 is not part of the integral if that's what you were worried about. The integration is from 0<x<1. z does not take part in the integral anyway. If you are interested in what happens to the result for ω=0 put it in and see.

Could somebody help me out ?
What have you tried so far?
How does this integral come up in the first place?
 
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  • #3
Simon Bridge said:
I'm more concerned with what ##x^\prime## represents.
It's not x', it's xl (x raised to the power of l).
 
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  • #4
Thanks for response.

1. Yes, I mean [itex] \lim_{k\rightarrow 0}(\omega+ik)[/itex].

2. As pointed out by D H, it is [itex]x^l[/itex], x raised to the power of l.

3. Initially, I tried to do it by this formula, [itex]\frac{1}{\omega+i0_+-x^m} = \mathcal{P}\left(\frac{1}{\omega-x^m}\right)-i\pi \delta(\omega - x^m)[/itex], with [itex]\delta(x)[/itex] denoting the Dirac function and [itex]\mathcal{P}[/itex] indicating the principal value. The imaginary part can thus be easily worked out. But I do not know how to handle the real part (i.e., the principal value part), which is supposed to contain a singularity at [itex]x^m = \omega[/itex].

4. I have also tried to make a change of variable, [itex]x^m \rightarrow y[/itex]. In terms of [itex]y[/itex], the integral becomes something like [itex]\int dy ~ \frac{y^{\nu}}{z-y}[/itex], with [itex]\nu = l/m<1[itex] (assumption). However, the same problem exists.

Then, how to do the principal value?
 
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  • #5
Have you tried rationalizing the denominator or converting it to a complex exponential?
have you tried u=z-x^m ... of course this makes u a complex number...
 

1. How do I find the limits of integration for a given integral?

The limits of integration can be found by looking at the function being integrated and the boundaries of the region being integrated over. For example, if the function is only defined over the interval [a,b], then the limits of integration would be a and b.

2. What are some common integration techniques?

Some common integration techniques include substitution, integration by parts, and partial fractions. Other techniques such as trigonometric substitution and using tables of integrals can also be used in certain cases.

3. How can I check my answer after integrating?

You can check your answer by taking the derivative of the integrated function and comparing it to the original function. If they are the same, then your answer is correct.

4. Can integrals be solved using software or do I have to do them by hand?

Integrals can be solved using software such as calculator or computer programs, but it is important to have a basic understanding of integration techniques in order to use these tools effectively.

5. Are there any tips for making integration easier?

Some tips for making integration easier include practicing and becoming familiar with common integration techniques, breaking down the integral into smaller parts, and using symmetry and other properties of the function to simplify the integral.

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