PHYS20672 Summary 6

Fourier Transforms and delta functions

1. The Fourier transform of $f(x)$ is $$F(k)=\frac 1 {\sqrt{2\pi}}\int_{-\infty}^\infty \e^{-i k x} f(x) \d x.$$
2. The normalisation, and the sign of $i$ in the exponential are conventions, and should be checked when other sources are used.
3. In physics $x$ usually represents space and $k$, wavenumber (or momentum over "hbar"). Where the initial variable is time $t$, the conjugate variable is angular frequency $\omega$, and the kernel $\e^{i \omega t}$is often used.
4. The inverse of the transformation is $$f(x)=\frac 1 {\sqrt{2\pi}}\int_{-\infty}^\infty \e^{i k x} F(k) \d k.$$
5. The Fourier transform of a Gaussian $\e^{-x^2/(2a^2)}$ is another Gaussian $a\e^{-k^2a^2/2}$.
6. The delta function is defined by the sifting property $\int_a^b f(x')\delta(x-x')\d x'=f(x)$ if $a < x < b$ and 0 otherwise.
7. A delta sequence of functions $\phi_n(x)$ has the property $\lim_{n\to\infty}\int_{-\infty}^\infty \phi_n(x')f(x')\d x'=f(0)$. An example is the "top hat" functions $\phi_n(x)=n$ for $-1/(2n) < x < 1/(2n)$ and 0 otherwise.
8. $\frac \kappa \pi$sinc$(\kappa x)$ is a delta sequence (parameter $\kappa$).
9. The identity $$f(x)=\frac 1 {\sqrt{2\pi}}\int_{-\infty}^\infty \e^{i k x} \left( \frac 1 {\sqrt{2\pi}}\int_{-\infty}^\infty \e^{-i k x'} f(x') \d x'\right)\d k$$ implies $$\frac 1 {2\pi}\int_{-\infty}^\infty \e^{i k (x-x')}\d k=\delta(x-x'),$$ which follows from the previous point (taking the integration limits to be $\pm \kappa$).
10. Parseval's theorem: $$\int_{-\infty}^\infty f^*(x)g(x)\d x=\int_{-\infty}^\infty F^*(k)G(k)\d k$$
11. Convolution: If $h(x)$ is the convolution of $f(x)$ and $g(x)$: $h(x)=f(x)*g(x)\equiv \int_{-\infty}^\infty f(y)*g(x-y)\d y$ then $H(k)=\sqrt{2\pi}F(k)G(k)$.