Commit 5a295407 authored by Reiner Zorn's avatar Reiner Zorn

added 'Kohlrausch' example

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......@@ -100,7 +100,7 @@ Although best effort is taken to ensure that the relations are correct no effort
Because this is a `dynamic' document there will often be the situation that sections are unfinished, or planned but due to workload reasons cannot be implemented immediately. In that case the \bb\ mark (defined as \verb|\bb|) will be used, sometimes extended by a short comment (defined as \verb|\note{}|).
This collection is published on a git repository. The reason for this choice is that it is possible to publish the \LaTeX\ source with full quality, instead of the reduced formula markup used in wikis. Admittedly, it makes collaboration a bit more complicated. If you want to add or change anything you can do this in the \LaTeX\ code and send a pull request. You can also just raise an issue like ``Please add formula (\dots) from reference [\dots]'' and I will include that if it is appropriate.
This collection is published on a git repository. The reason for this choice is that it is possible to publish the \LaTeX\ source with full quality, instead of the reduced formula markup used in wikis. Admittedly, it makes collaboration a bit more complicated. If you want to add or change anything you can do this in the \LaTeX\ code and send a pull request. You can also just raise an issue like ``Please add formula (\dots) from reference [\dots]'' and I will include that if it is appropriate. Finally, I would be glad if errors are reported in that way!
\subsection{Notation}
In most cases the notation here will follow the standard notation used in publications about hypergeometric functions. The use of $i$ as index will be avoided. Nevertheless $\i=\sqrt{-1}$ is non-italic for distinction and so are $\d$ and $\e$ in their special meanings. Real and imaginary part are denoted by (old-school) $\Re$ and $\Im$, respectively. A special notation was chosen for the standard `discriminants' constructed from the parameters of the Meijer $G$ function and the Fox $H$ function. They are written in sans s\'erif instead of the standard Greek letters as $\salpha, \sbeta \dots$ (\LaTeX: \verb|\salpha|\dots) to avoid confusion with ordinary symbols.
......@@ -634,7 +634,7 @@ The numerical calculation of the contour integral defining the Fox $H$ function
\end{eqm}
for $z>0$. If it is known that all parameters and the argument are real only the real part has to be calculated.
\subsection{Complex argument}
\subsection{Complex argument}\label{complex}
The Fox $H$ function of a complex argument $ z = x \exp (\i\phi) $ ($x \in \mathbb{R}^+$, $\phi \in \left]-\pi,+\pi\right]$) can be split into a real and an imaginary part:
\begin{eqm}
\Hpqmn =
......@@ -806,9 +806,9 @@ Laplace transform of the Fox $H$ function~\cite{Ma10} (valid on conditions state
\begin{eqnarray}
&& \Lap{ t^\alpha \H{p,q}{m,n}{\(a_1,A_1\)\dots\(a_p,A_p\)}{\(b_1,B_1\)\dots\(b_q,B_q\)}{c t^\beta} } = \nonumber\\
&& \hspace{3cm} = { 1 \over s^{1+\alpha} }
\H{p+1,q}{m,n+1}{\( -\alpha,\beta \), \(a_1,A_1\)\dots\(a_p,A_p\)}{\(b_1,B_1\)\dots\(b_q,B_q\)}{{ c \over s^\beta}} \\
\H{p+1,q}{m,n+1}{\( -\alpha,\beta \), \(a_1,A_1\)\dots\(a_p,A_p\)}{\(b_1,B_1\)\dots\(b_q,B_q\)}{{ c \over s^\beta}} \label{lap1}\\
&& \hspace{3cm} = { 1 \over s^{1+\alpha} }
\H{q,p+1}{n+1,m}{\(1-b_1,B_1\)\dots\(1-b_q,B_q\)}{\( 1+\alpha,\beta \), \(1-a_1,A_1\)\dots\(1-a_p,A_p\)}{{ s^\beta \over c }}
\H{q,p+1}{n+1,m}{\(1-b_1,B_1\)\dots\(1-b_q,B_q\)}{\( 1+\alpha,\beta \), \(1-a_1,A_1\)\dots\(1-a_p,A_p\)}{{ s^\beta \over c }} \label{lap2}
\end{eqnarray}
Inverse Laplace transform of the Fox $H$ function~\cite{Ma10} (valid on conditions stated therein):
\begin{eqnarray}
......@@ -821,6 +821,8 @@ Inverse Laplace transform of the Fox $H$ function~\cite{Ma10} (valid on conditio
Only the classes of the Fox $H$ functions and the Meijer $G$ functions are closed with respect to the inverse Laplace transform (without reciprocal arguments).
\subsection{Fourier Transform}
In general the (one sided) Fourier transform of the functions discussed here can be obtained by $s=\i\omega$ in the relations of the previous section. But this requires the calculation of the resulting hypergeometric function for an imaginary argument. For applications it may be interesting to convert these relations into expressions containing real arguments only.
Fourier Transform of the generalised hypergeometric function:
\begin{eqnarray}\label{ftf}
&& \Fou{\F{p}{q}{a_1\dots a_p}{b_1\dots b_q}{ct}} =
......@@ -852,16 +854,44 @@ Fourier Transform of the generalised hypergeometric function:
\end{eqs}
\end{quote}
Fourier transform of the generalised hypergeometric function multiplied with a power of its argument ($\alpha>-1$):
\bb
Fourier transform of the Fox $H$ function:
\bb
Fourier transform of the Fox $H$ function (for $\omega>0$, $c>0$):
\begin{eqnarray}
&& \Fou{ \H{p,q}{m,n}{\(a_1,A_1\)\dots\(a_p,A_p\)}{\(b_1,B_1\)\dots\(b_q,B_q\)}{ct} } = \nonumber\\
&& \quad = {\pi\over\omega} \H{p+2,q+1}{m,n+1}{(0,1),\(a_1,A_1\)\dots\(a_p,A_p\),(0,1/2)}{\(b_1,B_1\)\dots\(b_q,B_q\),(0,1/2)}{{c\over\omega}} \nonumber\\
&& \qquad - {\pi\over\omega}
\H{p+2,q+1}{m,n+1}{(0,1),\(a_1,A_1\)\dots\(a_p,A_p\),(1/2,1/2)}{\(b_1,B_1\)\dots\(b_q,B_q\),(1/2,1/2)}{{c\over\omega}} \i \label{ft1} \\
&& \quad = - {\pi\over\omega} \H{q+1,p+2}{n+1,m}{\(1-b_1,B_1\)\dots\(1-b_q,B_q\),(0,1/2)}{(1,1),\(1-a_1,A_1\)\dots\(1-a_p,A_p\),(0,1/2)}{{\omega\over c}} \nonumber\\
&& \qquad - {\pi\over\omega}
\H{q+1,p+2}{n+1,m}{\(1-b_1,B_1\)\dots\(1-b_q,B_q\),(1/2,1/2)}{(1,1),\(1-a_1,A_1\)\dots\(1-a_p,A_p\),(1/2,1/2)}{{\omega\over c}} \i \label{ft2}
\end{eqnarray}
More variants can be derived by combining equations \eq{lap1}, \eq{lap2} with the variants in equations \eq{complp}, \eq{compln} in different ways. More general relations can be obtained by setting $\alpha\ne0$ and/or $\beta\ne1$ in \eq{lap1} and \eq{lap2}.
The Fourier transform of the Fox-Wright function can only be expressed as Fox-Wright function of an imaginary argument (formula~\eq{lapfw} with $s=\i\omega$) or as Fox $H$ function using the above formula. The Fourier transform of the Meijer $G$ function can probably be expressed similarly to formula~\eq{ftf} via the duplication formula for $H$.
\begin{quote}
{\bf Example:} Calculate the real part of the Fourier transform of the Kohlrausch function $ \phi(t) = \exp\( - (t/\tau)^\beta \) $.
\[
\phi(t) =
\H{0,1}{1,0}{-}{(0,1)}{ \({t\over\tau}\)^\beta } =
{1\over\beta} \H{0,1}{1,0}{-}{(0,1/\beta)}{ {t\over\tau} }
\]
Applying \eq{ft2} with $c=1/\tau$ immediately gives the Fourier transform in terms of real Fox $H$ functions:
\[
\Fou{\phi(t)} =
- {\pi\over\beta} {1\over\omega} \H{2,2}{1,1}{(1,1/\beta),(0,1/2)}{(1,1),(0,1/2)}{\omega\tau}
- {\pi\over\beta} {1\over\omega} \H{2,2}{1,1}{(1,1/\beta),(1/2,1/2)}{(1,1),(1/2,1/2)}{\omega\tau} \i
\]
As a check of correctness the real part may be evaluated by the second calculable representation, \eq{calc2}:
\begin{eqnarray*}
\Re\( \Fou{\phi(t)} \) &=& - {\pi\over\beta} {1\over\omega}
\sum_{k=0}^{\infty} {\Gamma(1+\beta k) \over \Gamma(\beta k / 2) \Gamma(1-\beta k /2)}
{ (-1)^k \beta (\omega\tau)^{-\beta k} \over k! } \\
&=& \sum_{k=0}^{\infty}
(-1)^{k+1} {\Gamma(1+\beta k) \over k!} \sin{\pi\beta k \over 2} (\omega\tau)^{-\beta k -1}
\end{eqnarray*}
The result agrees with equations (13,14) in \Ref{Wu12}. (Note, that there implicitly $\tau=1$ was set and the sum equally can start at $k=1$ because the $k=0$ term vanishes.) Interestingly, the other series expansion in \Ref{Wu12} corresponds to the first calculable representation, \eq{calc1}, which is not supposed to converge for $\beta<1$ (being the case for most practical use) but still may have a purpose in the numerical calculation as an asymptotic series.
\end{quote}
\bibliographystyle{rz}
\bibliography{hypergeom}
......
......@@ -155,3 +155,16 @@ year = {2004},
PAGES={1},
YEAR={2000} }
@article{Wu12,
volume={5},
ISSN={1999-4893},
url={http://dx.doi.org/10.3390/a5040604},
DOI={10.3390/a5040604},
number={4},
journal={Algorithms},
publisher={MDPI AG},
author={Wuttke, Joachim},
year={2012},
month={Nov},
pages={604–628}
}
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