# Fourier series proof

#### Poirot

##### Banned
Let f(k) be the fourier transform of F(x). Prove that the fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the fourier transform is defined to have a factor of 1/2pi.

#### CaptainBlack

##### Well-known member
Let f(k) be the fourier transform of F(x). Prove that the fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the fourier transform is defined to have a factor of 1/2pi.
The particular form of the FT you are using is not that important, the basic idea is that:

$\mathfrak{F} \big[ F(ax) \big] (k) = \int_{-\infty}^{\infty} F(ax) e^{-kx{\rm{i}}}\; dx$

Now make the change of variable $$x^{*}=ax$$ and the result drops out.

CB

#### CaptainBlack

##### Well-known member
Let f(k) be the fourier transform of F(x). Prove that the fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the fourier transform is defined to have a factor of 1/2pi.
There is confusion here about the thread title (Fourier series) and content (Fourier transform). I have answered for the FT, is that what you intended?

CB