Sin/cos integrals multiplying results (fourier transform).

[binbagsss]

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

Where F(k)=$^{\infty}_{\infty}$$\int$f(x)exp$^{-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$\pi$x)=f(x)

Where F(k)=$^{\infty}_{\infty}$$\int$f(x)exp$^{-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.