# integral transform

A generic integral transform takes the form

 $F(p)=\int_{\alpha}^{\beta}K(p,t)f(t)dt,$

with $p$ being the transform parameter.

Note that the transform takes a function $f(t)$ and produces a new function $F(p)$.

The function $K(p,t)$ is called the kernel of the transform. The kernel of an integral transform, along with the limits (http://planetmath.org/DefiniteIntegral) $\alpha$ and $\beta$, distinguish a particular integral transform from another.

## Examples

•  $\displaystyle\alpha=0,\;\beta=\infty,\;K(p,t)=e^{-pt},$ $\displaystyle F(p)=\int\limits_{0}^{\infty}e^{-pt}f(t)dt.$
• Laplace-Carson transform

 $\displaystyle\alpha=0,\;\beta=\infty,\;K(p,t)=pe^{-pt},$ $\displaystyle F(p)=\int\limits_{0}^{\infty}pe^{-pt}f(t)dt.$
•  $\displaystyle\alpha=-\infty,\;\beta=\infty,\;K(p,t)=\frac{1}{\sqrt{2\pi}}e^{-% ipt},$ $\displaystyle F(p)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}e^{-ipt}% f(t)dt.$
Title integral transform IntegralTransform 2013-03-22 12:34:03 2013-03-22 12:34:03 PrimeFan (13766) PrimeFan (13766) 10 PrimeFan (13766) Definition msc 65R10 ContourIntegral GroupHomomorphism kernel transform parameter