integral transformIntegralTransform2002-04-05 17:14:522008-09-03 18:59:40Definitionkerneltransform parameter\usepackage{amssymb}
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\PMlinkescapeword{generic}
\PMlinkescapeword{transform}
A generic \emph{integral transform} takes the form $$F(p) = \int_\alpha^\beta K(p, t) f(t)dt,$$ with $p$ being the {\em 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 \emph{kernel} of the transform. The kernel of an integral transform, along with the \PMlinkname{limits}{DefiniteIntegral} $\alpha$ and $\beta$, distinguish a particular integral transform from another.
\subsection*{Examples}
\begin{itemize}
\item Laplace transform
\begin{gather*}
\alpha = 0,\; \beta = \infty,\; K(p,t) = e^{-pt},\\
F(p) = \int\limits_0^\infty e^{-pt} f(t) dt.
\end{gather*}
\item Laplace-Carson transform
\begin{gather*}
\alpha = 0,\; \beta = \infty,\; K(p,t) = pe^{-pt},\\
F(p) = \int\limits_0^\infty pe^{-pt} f(t) dt.
\end{gather*}
\item Fourier transform
\begin{gather*}
\alpha = -\infty,\; \beta = \infty,\; K(p,t) = \frac{1}{\sqrt{2\pi}}e^{-ipt},\\
F(p) = \frac{1}{\sqrt{2\pi}}
\int\limits_{-\infty}^\infty e^{-ipt} f(t) dt.
\end{gather*}
\end{itemize}