Sin/cos integrals multiplying results (fourier transform).

[binbagsss]

Okay, I am trying to determine the Fourier transform of cos (2\pix)=f(x)

Where F(k)=^{\infty}_{\infty}\intf(x)exp^{-ikx} dx,

So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as:

^{\infty}_{-\infty}\int cos(nx)cos(mx) dx =\pi if m=n≠0
= 2\pi if m=n=0
=0 if m≠n

- But these are only defined for limits \pm\pi.
So my question is , what are these results for \pm\infty.

Is there a obvious natural extension?

Many thanks for any assistance !

 

[LCKurtz]

Okay, I am trying to determine the Fourier transform of cos (2\pix)=f(x)

Where F(k)=^{\infty}_{\infty}\intf(x)exp^{-ikx} dx,

So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as:

^{\infty}_{-\infty}\int cos(nx)cos(mx) dx =\pi if m=n≠0
= 2\pi if m=n=0
=0 if m≠n

- But these are only defined for limits \pm\pi.
So my question is , what are these results for \pm\infty.

Is there a obvious natural extension?

Many thanks for any assistance !

One of the requirements for a Fourier transform of a function $f$ to exist is$$
\int_{-\infty}^\infty |f(x)|~dx$$converge. Sines and cosines don't satisfy that. Apparently there is some sense in which it can be expressed with delta functions.