# Eikonal Blog

## 2010.01.04

### Fourier transform

Filed under: mathematics — Tags: , — sandokan65 @ 20:31

Definition:

• Direct: $F(p) = {\bf F}[f](p) :\equiv \int_{-\infty}^{+\infty} \frac{dt}{(2\pi)^{\mu}} e^{ipt} f(t)$,
• Inverse: ${\bf F}^{-1}[g](t) :\equiv \int_{-\infty}^{+\infty} \frac{dp}{(2\pi)^{1-\mu}} e^{-ipt} g(p)$.

Here some author use $\mu=1$, and some authors $\mu=\frac12$ (e.g. Mitrinović)

Properties:

• ${\bf F}[f(t+a)](p) = e^{iap} F(p)$
• ${\bf F}[e^{iat}f(t)](p) = F(p+a)$
• if $\lim_{t\rightarrow \pm\infty} |f(t)| = 0$, then ${\bf F}[f'(t)](p) = -i p F(p)$
• ${\bf F}[\int_{{\Bbb R}}f(t-u)g(u)\frac{du}{(2\pi)^{\mu}}](p) = F(p) G(p)$

## Table of Fourier transformations

$f(t)$ $F(p)={\bf F}[f](p)$
$\frac{\sin(at)}{t}$ $\frac{1}{2}\sqrt{\frac{\pi}{2}}(\frac{p+a}{|p+a|} - \frac{p-a}{|p-a|})$
$exp(-a t^2)$ $\frac{1}{\sqrt{2a}} exp(-\frac{p^2}{4\pi})$
$\cos(at^2)$ $\frac{1}{\sqrt{2a}} \cos(\frac{p^2}{4a}-\frac{\pi}{4})$
$\sin(at^2)$ $\frac{1}{\sqrt{2a}} \sin(\frac{p^2}{4a}+\frac{\pi}{4})$
$|t|^{-a}$ ($0) \$ $\sqrt{\frac{2}{\pi}} |p|^{1-a} \Gamma(1-a) \sin(\frac{\pi a}{2})$
$(1+t^2)^{-1}$ $\sqrt{\frac{\pi}{2}} e^{-|p|}$

## Sources

• 1) D.S. Mitrinović, J.D. Kečkić: “Jednačine Matematičke Fizike”; 2nd edition; Građevinska Knjiga, Beograd 1978.

### 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 $n$).

### Trinomial equation related to Mellin transform

Filed under: mathematics — Tags: , , , — sandokan65 @ 20:08

The algebraic equation $y^n + x y^p -1 = 0$ for ($n>p$) has following series solutions:

$y_0(x) = \frac1{n} \sum_{r=0}^\infty \frac{(-)^r}{r!} \frac{\Gamma(\frac{1+pr}{n}) x^r}{\Gamma(\frac{1+pr}{n}+1-r)}.$

Other $n-1$ solutions are given by the complex rotations as $y_k(x) = \omega^k y_0(\omega^{pk} x)$ where $\omega:\equiv e^{i\frac{2\pi}{n}}$, $k=\overline{0,n-1}$.

The solutions satisfy following ODE:

$(-d_x)^n y(x) = p (-x d_x -n) y(x)$.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt