Representation of Delta Function

[poonintoon]

Hopefully people are still prowling the forums this close to christmas :)

I want to show that sin(ax)/x is a representation of a delta function in the limit a->infinty i.e
1) It equals 0 unless x=0
2) integrated from plus minus infinity it equals 1 and
3) multiplying by an arbitrary function f(x) and integrating gives f(0)

but I cannot show any of these. I have tried series representation, writing out as exponentials, looking up definite integrals etc but cannot make any headway.

Cheers

 

[hunt_mat]

The integral of your representation can be found by employing residue calculus. I believe you have a factor of \pi missing.

 

[poonintoon]

Hi,

Thanks, you are right about the pi factor. But what if the function f(x) was say x then there would not be a pole and residue calculus wouldn't make sense would it?

 

[Avodyne]

Not every function is allowed as a test function. See

http://en.wikipedia.org/wiki/Distribution_(mathematics )