Integral from 0 to ∞ with singularity at x=0

In summary, the conversation is about an integral that is causing difficulties. Different approaches have been tried, including contour integration and replacing the exponential with its Taylor sum. The even terms have been successfully integrated, but there is uncertainty about how to handle the odd terms. The integral has been rewritten in a form that requires integrals of a specific type, but the speaker is still unable to solve it. There is a question about whether the integration is over the real axis, but it is clarified that x is real and the direction of integration should not affect the result. The idea of the gamma-function is brought up as a possible solution.
  • #1
RedSonja
21
0
Here's an integral that is currently giving me grey hairs:

[itex]\int_0^{\infty} \frac{1}{x} \exp(i \frac{k}{x}(a-c \cos(\theta + wx))) dx[/itex]

I've tried different approaches like contour integration around [itex] x=0 [/itex] and replacing the exponential by its Taylor sum to have:

[itex]\int_0^{\infty} \sum_{n=0}^{\infty} \frac{1}{x^{n+1}\;n!} (i k (a-c \cos(\theta + wx)))^n dx[/itex]

I can do the integrals of the even terms by [itex]\int_0^{\infty} = \frac{1}{2}\int_{-\infty}^{\infty}[/itex] and residues, but I don't know how to handle the odd terms.

Going to sum-extremes I've rewritten the integral to a form where I only need to do integrals of the type

[itex]\int_0^{\infty} \frac{1}{x^{n+1}} e^{imwx} dx [/itex]

with m a positive or negative integer, but even here I'm stuck. Please help!
 
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  • #2
Maybe this is a stupid question, but is the integration over the real axis?
 
  • #3
rollingstein said:
Maybe this is a stupid question, but is the integration over the real axis?

Yes, x is real. But since the integrand goes to zero for [itex]x\rightarrow \infty [/itex] the direction of integration in the complex plane shouldn't alter the integral...
 
  • #4
Hmm.
Is it possible that the latter of the three integrals is simply the gamma-function?
 

1. What is the meaning of an integral from 0 to ∞ with a singularity at x=0?

An integral from 0 to ∞ with a singularity at x=0 is a mathematical expression that represents the area under a curve from 0 to infinity, where the function has a discontinuity or "singularity" at x=0. This means that the function is undefined at x=0, and the integral must be evaluated using special techniques.

2. Why is there a singularity at x=0 in this integral?

The singularity at x=0 in this integral is typically caused by a function that is undefined at that point. This could be due to a division by zero, logarithmic or trigonometric functions with asymptotes at x=0, or other mathematical reasons. It is important to note that singularities are not always present in integrals, but when they are, they require special attention to solve.

3. Can an integral from 0 to ∞ with a singularity at x=0 be solved using traditional integration techniques?

In most cases, no. Traditional integration techniques such as the fundamental theorem of calculus and integration by parts rely on the function being continuous and well-defined at all points. With a singularity at x=0, these techniques cannot be used. However, there are specialized techniques such as Cauchy's principal value and contour integration that can be used to solve integrals with singularities.

4. What is the physical significance of an integral from 0 to ∞ with a singularity at x=0?

The physical significance of such an integral depends on the specific function being integrated. In some cases, it may represent a physical quantity such as the total energy of a system or the area under a curve in a real-world scenario. In other cases, it may have no physical interpretation and is simply a mathematical concept used to solve problems in calculus or other branches of mathematics.

5. How does one handle the singularity at x=0 when evaluating this integral?

Handling the singularity at x=0 depends on the specific integral and function being evaluated. In general, special techniques must be used to "remove" the singularity, such as taking a limit as x approaches 0 or using a substitution to transform the integral into a form that can be solved using traditional techniques. It is important to carefully analyze the function and choose the appropriate method to handle the singularity in order to obtain a correct solution.

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