Help with double integral of exp(ixy)

In summary, the double integral \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {e^{ixy} dxdy = 2\pi} when x and y are real. This value can be derived by converting the integral to polar coordinates and using a Bessel J function, and then taking the limit as the bounds approach infinity. Alternatively, the value can also be derived using the Laplace transform.
  • #1
lemma28
18
1
Please help me with folllowing double integral


[tex]\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {e^{ixy} dxdy = 2\pi}}[/tex]

(x,y, real)

It came out analyzing the relation between DiracDelta and the Fourier Transform formula. (it's the reason why insert the constant 1/sqrt(2pi) in the Fourier transform formula to be consistent with the diracdelta definition).
I know that it's value is 2pi. But I'd like to see how to actually calculate it. (Possibly in some elegant way...)

There must be some tricky "magic" based on symmetry consideration to reduce the double integral to the length of a unit circle. But I can't find it.

Thanks
 
Physics news on Phys.org
  • #2
You could try converting the integral to polar coordinates,

[tex]\int_{0}^{2\pi}d\varphi \int_0^{\infty}e^{i\frac{r^2}{2}\sin 2\varphi}rdr[/tex]

then let [itex]z = r^2/2[/itex] and [itex]\theta = 2\varphi[/itex]. Your angular integral should then look like a Bessel J function. http://www.math.sfu.ca/~cbm/aands/page_360.htm look at relation 9.1.21.
 
Last edited:
  • #3
If you change the bounds to some finite range, say -a<x<a and -b<y<b, then it shouldn't be too hard to show that the integral reduces to:

[tex]2 \int_{-ab}^{ab} \frac{\sin(u)}{u} du [/tex]

Then you can take the limit as a,b-> infinity. The improper integral of sin(x)/x is known to be pi, and the easiest way to derive this is probably using a laplace transform.
 
  • #4
Thanks. I got it!
 

1. What is a double integral?

A double integral is a type of mathematical integration that involves integrating a function of two variables over a two-dimensional region. It can be thought of as the area under a surface in three-dimensional space.

2. What does the function exp(ixy) mean?

The function exp(ixy) is the complex exponential function, where i is the imaginary unit and x and y are real numbers. It is commonly used in mathematics and physics to represent oscillatory behavior.

3. How do you solve a double integral of exp(ixy)?

The process of solving a double integral of exp(ixy) involves using techniques such as substitution, integration by parts, and trigonometric identities. It may also require the use of advanced mathematical concepts such as contour integration or Fourier transforms.

4. What are the applications of double integrals of exp(ixy)?

Double integrals of exp(ixy) have various applications in mathematics, physics, and engineering. They are often used to solve problems involving wave propagation, quantum mechanics, and signal processing.

5. Are there any tips for solving a double integral of exp(ixy)?

Some tips for solving a double integral of exp(ixy) include carefully choosing the limits of integration, using symmetry to simplify the integral, and breaking down the integral into smaller parts. It is also helpful to have a strong understanding of complex numbers and integration techniques.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
809
Replies
7
Views
3K
Replies
2
Views
4K
Replies
1
Views
697
Replies
1
Views
2K
  • Sticky
  • Topology and Analysis
Replies
9
Views
5K
Replies
1
Views
609
  • Calculus
Replies
2
Views
2K
Replies
8
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
996
Back
Top