Evaluating Complex Integral: 2$\int_0^{\infty} cos(-Ax) e^{-Bx^2} dx$

In summary, the conversation discusses the evaluation of the integral \int_{-\infty}^{\infty} e^{-Bx^2} e^{-iAx} dx. It is suggested to split the integral using Euler's identity and then evaluate the imaginary and real parts separately. The real part is found to be 2 \int_0^{\infty} cos(-Ax) e^{-Bx^2} dx, which proves difficult to integrate. A suggestion is made to complete the square in the exponential and then use complex analysis techniques, such as choosing a loop in the complex plane to integrate around. However, it is mentioned that the person has no knowledge of complex analysis. After some discussion, it is found that
  • #1

quasar987

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Now I have to evaluate

[tex]\int_{-\infty}^{\infty} e^{-Bx^2} e^{-iAx} dx[/tex]

Splitting it in two using Euler's identity show that the imaginary part is 0 (cuz integrand is odd). Remains the real part

[tex]2 \int_0^{\infty} cos(-Ax) e^{-Bx^2} dx[/tex]

for which integration by parts leads nowhere.
 
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  • #2
I recommend completing the square in the exponential then suitably choosing a loop around which to integrate in the complex plane.
 
  • #3
I should have mentionned that I have no knowledge of complex analysis whatesoever. I don't know anything about residues, integration in the complex plane, and stuff like that.
 
  • #4
Since a complex number appears in the exponential you are at least familiar with some of the basics. I still recommend completing the square in the exponential. It should provide some illumination. :)
 
  • #5
I love ilumination. Let me try just that. :tongue2:
 
  • #6
Is it just [itex]\sqrt{\pi / B}[/itex]?

After completing the square, I'm left with

[tex]\mbox{exp}(-A^2 / 4B) \int \mbox{exp}(-B(x+Ai/2B)^2) dx[/tex]

And so with substitution y = x+Ai/2B, I get the integral

[tex]\mbox{exp}(-A^2 / 4B) \int_{-\infty}^{\infty} \mbox{exp}(-By^2) dy[/tex]

which is [itex]\sqrt{\pi / B}[/itex] as pointed out by another thread by Tom Mattson.

Is this valid with complex too?
 
Last edited:
  • #7
quasar,

Way to go!
 

1. What is a complex integral?

A complex integral is a mathematical concept in which a function is integrated over a complex domain. Unlike real integrals, which are calculated along a single axis, complex integrals are calculated along a complex curve in the complex plane.

2. What does it mean to evaluate a complex integral?

Evaluating a complex integral means finding the numerical value of the integral. This involves using mathematical techniques such as integration methods and theorems to simplify the integral and solve for its value.

3. How do I evaluate an integral with complex numbers?

To evaluate an integral with complex numbers, you can use the same techniques as you would for real integrals, such as substitution, integration by parts, and partial fractions. However, you must also take into account the properties of complex numbers, such as conjugates, in order to simplify the integral.

4. What is the purpose of the integral in the given equation?

In this equation, the integral is used to find the area under the curve of the function cos(-Ax) e^{-Bx^2}. It is a way to calculate the total value of the function over a given interval.

5. Can the given integral be solved analytically?

Yes, the given integral can be solved analytically using integration techniques and theorems. However, it may involve complex numbers and may require advanced mathematical knowledge to solve.

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