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

• Laplace transform

  $$\alpha =0,\beta =\mathrm{\infty },K(p,t)=e^{-pt},$$

  $$F(p)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-pt}f(t)𝑑t.$$

• Laplace-Carson transform

  $$\alpha =0,\beta =\mathrm{\infty },K(p,t)=p{e}^{-pt},$$

  $$F(p)=\underset{0}{\overset{\mathrm{\infty }}{\int }}p{e}^{-pt}f(t)𝑑t.$$

• Fourier transform

  $$\alpha =-\mathrm{\infty },\beta =\mathrm{\infty },K(p,t)=\frac{1}{\sqrt{2\pi }}{e}^{-ipt},$$

  $$F(p)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-ipt}f(t)𝑑t.$$