How Does the Integral Property Relate to the Dirac Delta Function?

[Apteronotus]

Hi everyone,

Can anyone show me how the property
[tex]\frac{1}{2\pi} \int ^{\infty} _{-\infty} e^{i\omega x}d\omega= \delta(x)[/tex]
holds.

Thanks,

 

[g_edgar]

Do you know about Schwartz distribution theory? Generalized functions? Because that is what you are writing about, not traditional calculus functions. Your equation MEANS...

[tex]
\frac{1}{2\pi}\int_{-\infty}^\infty e^{i\omega x} \phi(x)\,d\omega = \phi(x)
[/tex]

for all test functions [itex]\phi[/itex] from an appropriate class.

 

[Apteronotus]

edgar thanks for your reply.
It seems I've stumbled on something beyond my means.
I don't know anything about Schwartz distribution theory and a quick search on the net didnt help at all.

I thought the integral would be an easy calculus identity of sorts. Can you show me why the integral holds?