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%
% transformation.tex
%
% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule
%
\section{Laguerre Transformation
\label{laguerre:section:transformation}}
\begin{align}
L \left\{ f(x) \right\}
=
\tilde{f}_\alpha(n)
=
\int_0^\infty e^{-x} x^\alpha L_n^\alpha(x) f(x) dx
\label{laguerre:transformation}
\end{align}
\begin{align}
L^{-1} \left\{ \tilde{f}_\alpha(n) \right\}
=
f(x)
=
\sum_{n=0}^{\infty}
\begin{pmatrix}
n + \alpha \\
n
\end{pmatrix}^{-1}
\frac{1}{\Gamma(\alpha + 1)}
\tilde{f}_\alpha(n)
L_n^\alpha(x)
\label{laguerre:inverse_transformation}
\end{align}
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