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).
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