Solving Complex Gaussian Integral Problems

In summary, the conversation discusses the equation \int_{-\infty}^\infty e^{-ax^2}dx = \sqrt{\pi \over a}, and whether it holds for complex values of a. The participants conclude that it does hold for Re(a)>0 and that the equation relies on e^{-ax^2}\arrowvert_{\infty}\,=\,0. They also discuss using this equation to find a Green function, but there is a discrepancy as the integral formula used should not work for Re(a)<0.
  • #1
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4

Homework Statement


We know that
[tex]\int_{-\infty}^\infty e^{-ax^2}dx = \sqrt{\pi \over a}[/tex].

Does this hold even if [tex]a[/tex] is complex?


Homework Equations





The Attempt at a Solution


In the derivation of the above equation, I don't see any reason why we must assume that [tex]a[/tex] be real. So I think it does hold for complex [tex]a[/tex].
 
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  • #3
benorin said:
It holds for Re(a)>0.

Yes … it relies on [tex]e^{-ax^2}\arrowvert_{\infty}\,=\,0[/tex] .

If a = b + ic, then [tex]e^{-ax^2}=\,e^{-bx^2}e^{-icx^2}[/tex] , which is 0 if b > 0, and really wobbly if b ≤ 0. :smile:
 
  • #4
Great. Thank you for your replies.
 
  • #5
Hi again. I'd just like to make a remark that is bothering me. I created this thread because I was trying to find a Green function and arrived at
[tex]G(x,x',t) = {1\over{2\pi}}e^{im(x-x')\over{2t}}\int_{-\infty}^\infty e^{-{it\over{2m}}\left(p-{m(x-x')\over t}\right)^2}[/tex].
Please ignore everything except the integral.
When I naiively use the gaussian integral formula in my original post, I get the correct answer. But according to what you said, I should not be able to do this since (i.e. Re(a)<0) in this case. Can you see any reason for this?
 

1. What is the purpose of solving complex Gaussian integral problems?

The purpose of solving complex Gaussian integral problems is to determine the value of a complex Gaussian integral, which is a type of definite integral involving a complex-valued integrand. These integrals are important in various areas of mathematics and physics, and their solutions can provide insights into the behavior of complex functions.

2. What are some common techniques used to solve complex Gaussian integrals?

Some common techniques used to solve complex Gaussian integrals include the use of contour integration, substitution methods, and the application of the Cauchy integral theorem. These methods involve manipulating the integrand and utilizing properties of complex numbers to simplify the integral and find its solution.

3. Are there any special cases of complex Gaussian integrals that are easier to solve?

Yes, there are several special cases of complex Gaussian integrals that have simpler solutions. These include integrals with symmetric limits, integrals with certain types of rational functions as the integrand, and integrals with simple poles. In these cases, the integral can often be evaluated using basic integration techniques.

4. How can complex Gaussian integrals be applied in real-world problems?

Complex Gaussian integrals have a wide range of applications in various fields such as physics, engineering, and statistics. They can be used to calculate probabilities, solve differential equations, and model physical phenomena. For example, they are commonly used in quantum mechanics to determine the probability of a particle's location in space.

5. Are there any software programs that can assist with solving complex Gaussian integrals?

Yes, there are various software programs and online calculators that can assist with solving complex Gaussian integrals. Some popular options include Wolfram Alpha, Matlab, and Maple. These programs use advanced algorithms and numerical methods to quickly and accurately evaluate complex integrals.

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