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\begin{document}

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\section*{Mellin transform}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_the_exponential_function}{Of the exponential function}\dotfill \pageref*{of_the_exponential_function} \linebreak
\noindent\hyperlink{zeta_functions}{Zeta functions}\dotfill \pageref*{zeta_functions} \linebreak
\noindent\hyperlink{1LoopVacuumAmplitudes}{1-loop vacuum amplitudes}\dotfill \pageref*{1LoopVacuumAmplitudes} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{zeta_functions_2}{Zeta functions}\dotfill \pageref*{zeta_functions_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For $\tau \mapsto f(\tau)$ a suitably well-behaved [[function]] of a single [[variable]], its \emph{Mellin transform} is the function given by the [[integral]] expression

\begin{displaymath}
s \mapsto \int_0^\infty \tau^{s-1} f(\tau) \; d\tau
 \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{of_the_exponential_function}{}\subsubsection*{{Of the exponential function}}\label{of_the_exponential_function}

For the [[exponential function]] $\exp(-(-)A)$ the Mellin transform is proportional to the inverse [[power]] $A^{-(-)}$

\begin{displaymath}
\int_0^\infty \tau^{s-1} \exp(-\tau A) \,d\tau
  =
  (s-1)! \, A^{-s}
  \,.
\end{displaymath}
For $s = 1$ this has a [[complex number|complex]] analougue given by the [[Fourier transform]] of the [[Heaviside distribution]] (\href{Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution}{this example}).

In [[perturbative quantum field theory]], especially in the computation of [[propagators]] such as the [[Feynman propagator]], this formula is known as the \emph{[[Schwinger parameterization]]} for $A^{-s}$, leading to the ``[[worldline formalism]]''. See below at \emph{\hyperlink{1LoopVacuumAmplitudes}{1-Loop amplitudes}}.

\hypertarget{zeta_functions}{}\subsubsection*{{Zeta functions}}\label{zeta_functions}

A [[zeta function]]/[[L-function]] is the [[analytic continuation]] of the Mellin transform of the corresponding [[theta function]].

In particular it sends the [[Jacobi theta function]] to the (completed) [[Riemann zeta function]]:

\begin{displaymath}
\hat \zeta(s) = \int_0^\infty t^{s-1} \hat \theta(t) \, d t
\end{displaymath}
More generally, the Mellin transform appears as a stage in the expression of [[zeta functions]] as [[adelic integrals]] in [[Iwasawa-Tate theory]].

\hypertarget{1LoopVacuumAmplitudes}{}\subsubsection*{{1-loop vacuum amplitudes}}\label{1LoopVacuumAmplitudes}

[[1-loop vacuum amplitudes]] in [[quantum field theory]] are analytically continued Mellin transforms of [[partition functions]]. Here the parameter $\tau$ is called the \emph{[[Schwinger parameter]]} and the Mellin transform turns the [[worldline formalism]]-picture into the [[Feynman propagator]]-picture.

\begin{displaymath}
Tr H^{-s} = \int_0^\infty t^{s-1} Tr\, \exp(- t H) \, d t \,.
\end{displaymath}
\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\hypertarget{zeta_functions_2}{}\subsubsection*{{Zeta functions}}\label{zeta_functions_2}

\begin{itemize}%
\item [[zeta function of an elliptic differential operator]]


\item [[Riemann zeta function]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Laplace transform]]


\item [[Fourier transform]]


\item the generalization of Mellin transforms from [[automorphic forms]] to [[automorphic representation]] is the concept of [[automorphic L-function]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Mellin_transform}{Mellin transform}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Schwinger_parametrization}{Schwinger parameterzation}}


\item [[Joel Shapiro]], \emph{Schwinger trick and Feynman parameter}, 2007 (\href{https://www.physics.rutgers.edu/grad/615/lects/schwingertrick.pdf}{pdf})



\end{itemize}
For the [[adelic integral]]-version see at \emph{[[Iwasawa-Tate theory]]}.

For [[function fields]]:

\begin{itemize}%
\item [[David Goss]], \emph{A formal Mellin transform in the arithmetic of function fields}, Transactions of the AMS, volume 327, Number 2, October 1991 (\href{http://www.ams.org/journals/tran/1991-327-02/S0002-9947-1991-1041048-5/S0002-9947-1991-1041048-5.pdf}{pdf})

\end{itemize}
For the appearance in [[physics]] as integrals over [[Schwinger parameters]] producing [[Feynman propagators]] see

\begin{itemize}%
\item [[Joel Shapiro]], \emph{Schwinger trick and Feynman parameters} (\href{https://www.physics.rutgers.edu/grad/613/615lects/schwingertrick.pdf}{pdf})


\item Stefan Weinzierl, section 4.2.1 of \emph{Mathematical aspects of particle physics}, 2010 (\href{http://wwwthep.physik.uni-mainz.de/~stefanw/download/script_thep2.pdf}{pdf})



\end{itemize}
[[!redirects Mellin transforms]]

[[!redirects Schwinger parameter]] [[!redirects Schwinger parameters]]

[[!redirects Schwinger parameterization]] [[!redirects Schwinger parameterizations]]

[[!redirects Schwinger parameterisation]] [[!redirects Schwinger parameterisations]]



\end{document}