# Eikonal Blog

## 2010.01.04

### Integral transforms

Filed under: mathematics — Tags: — sandokan65 @ 20:29

General linear integral transformation:

$F(p) = {\bf T}[f](p) :\equiv \int_a^b K(p,t) f(t) dt$

Here we assume that this integral exist, $K$ is a fixed complex function, $p \in {\Bbb C}$ and $a,b\in{\Bbb R}$.

Specific examples:

• Laplace transformation: ${\bf L}$ defined by $a=0$, $b=+\infty$, $K(p,t)=e^{pt}$.
• Fourier transformation: ${\bf F}$ defined by $a=-\infty$, $b=+\infty$, $K(p,t)=\frac1{\sqrt{2\pi}}e^{ipt}$.
• Mellin transformation: ${\bf M}$ defined by $a=0$, $b=+\infty$, $K(p,t)=t^{p-1}$.
• Hankel transformation: ${\bf H}$ defined by $a=0$, $b=+\infty$, $K(p,t)=t \ J_n(pt)$ (a Bessel function of order $n$).