Integral of sinc(x)

  • Thread starter Sistine
  • Start date
  • #1
21
0

Homework Statement


I'm trying to prove the following definite integral of [tex]sinc(x)[/tex]

[tex]\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx=\pi[/tex]


Homework Equations





The Attempt at a Solution


I've tried power series expansions as well as trigonometric identities like

[tex]\frac{\cos 2x}{x}=\frac{\cos^2 x}{x}-\frac{\sin^2 x}{x}[/tex]

I also looked at techniques used to integrate the definite integral
[tex] \int_{-\infty}^{\infty}e^{-x^2}dx [/tex]

which I know is solved by double integration and changing to polar coordinates. However, this does not help me integrate [tex]sinc(x)[/tex].
 

Answers and Replies

  • #2
180
4
Well, I suppose you could do it by making a closed curve in the complex plane and using Caychy's theorem (and Jordan's lemma). There might be an easier way, but I can't think of any.
 
  • #3
Think about euler's formula and leibniz. A 'simple' proof can be made this way.
 
  • #4
1,851
7
[tex]\int_{0}^{\infty}\sin(x)\exp(-sx)dx=\frac{1}{1+s^2}[/tex]

Integrate both sides from s = 0 to infinity to obtain the result.
 
  • #6
sinc(x) = sin(x)
x
has no anti-derivative
 
  • #7
sinc(x) = sin(x)
x
has no anti-derivative

no elementary anti-derivative :wink:
 

Related Threads on Integral of sinc(x)

  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
1
Views
7K
Replies
9
Views
2K
Replies
3
Views
6K
  • Last Post
Replies
1
Views
14K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
4
Views
2K
Top