Math: Fourier Integral

We define the decomposition into plane waves of a function $f(x)$ of one variable $x$ by its Fourier transform $\tilde{f}(k)$,

$$\tilde{f}(k) := \int_{-\infty}^{\infty}dx {f}(x) e^{-ikx}, \quad f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk \tilde{f}(k) e^{ikx}.$$ (43)

Remarks:

1. In this lecture, we define the Fourier transform with the factor $1/2\pi$ as in (1.43). Some people define it symmetrically, i.e. $1/\sqrt{2\pi}$ in front of $f(x)$ and $\tilde{f}(k)$.

2. Remember the Minus signs in the $\exp$ functions.

We often use the Fourier transform $\tilde{f}({\bf k})$ in $d$ dimensions for a function $f({\bf x})$, ${\bf x}=(x_1,...,x_d)$,

$$\tilde{f}({\bf k}) := \int_{-\infty}^{\infty}d{\bf x} {f}({\bf x}) e^{-i{\bf kx}},\quad... ...c{1}{(2\pi)^d}\int_{-\infty}^{\infty}d{\bf k} \tilde{f}({\bf k}) e^{i{\bf kx}}.$$ (44)