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% transformation.tex 
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% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule
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\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}