Integral of e^(i x v) /(x^2 + a ^2) from -Infty to Infty

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In summary, the purpose of finding the integral of e^(i x v) /(x^2 + a ^2) from -Infty to Infty is to calculate the area under the curve of this complex-valued function. This can have various practical applications in physics, engineering, and mathematics. The integral can be solved using the method of residue integration, which involves finding the singularities of the function and using the Cauchy residue theorem. The singularities of this function are at x = ia and x = -ia, and other methods such as contour integration and series expansion can also be used to evaluate the integral. The exponential term e^(i x v) in the integral represents an oscillatory function that requires the use of
  • #1
kurushishraqi
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Hi!


Solving a problem in quantum mechanics I found this integral, but I have no idea how to solve it:

[tex]\int_{-\infty}^{ \infty} du \frac{e^{iuv}}{u^2+a^2}[/tex]

with [tex]a \in [/tex]Reals.

If somebody have an idea, it would be appreciated. Thanks!
 
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  • #2
Hmm... are you familiar with contour integration?
 
  • #3
Well, no. But with that info now I can solve my problem. Thanks a lot!
 

1. What is the purpose of finding the integral of e^(i x v) /(x^2 + a ^2) from -Infty to Infty?

The purpose of finding this integral is to solve for the area under the curve of the function e^(i x v) /(x^2 + a ^2) from negative infinity to positive infinity. This can be useful in various applications in physics, engineering, and mathematics.

2. How do you solve the integral of e^(i x v) /(x^2 + a ^2) from -Infty to Infty?

This integral can be solved using techniques from complex analysis, specifically the method of residue integration. It involves finding the residues of the function at its singularities and then using the Cauchy residue theorem to evaluate the integral.

3. What are the singularities of the function e^(i x v) /(x^2 + a ^2)?

The singularities of this function are at x = ia and x = -ia, where a is a real number. These are known as simple poles and are the only singularities of this function.

4. Can the integral of e^(i x v) /(x^2 + a ^2) from -Infty to Infty be evaluated using other methods?

Yes, there are other methods that can be used to evaluate this integral, such as contour integration and series expansion. However, the method of residue integration is the most efficient and accurate method for this particular integral.

5. What is the significance of the exponential term e^(i x v) in the integral of e^(i x v) /(x^2 + a ^2) from -Infty to Infty?

The exponential term e^(i x v) represents a complex-valued function that is oscillatory in nature. This adds a level of complexity to the integral and requires the use of complex analysis techniques to evaluate it. Additionally, the term e^(i x v) is often used in mathematical models to represent waves and vibrations.

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