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ModifiedMeijerG

modified Meijer G function

 Calling Sequence ModifiedMeijerG(as, bs, cs, ds, z)

Parameters

 as - list of the form [a1, ..., am]; first group of numerator $\mathrm{\Gamma }$ parameters bs - list of the form [b1, ..., bn]; first group of denominator $\mathrm{\Gamma }$ parameters cs - list of the form [c1, ..., cp]; second group of numerator $\mathrm{\Gamma }$ parameters ds - list of the form [d1, ..., dq]; second group of denominator $\mathrm{\Gamma }$ parameters z - expression

Description

Important: The ModifiedMeijerG command has been deprecated.  Use the superseding command MeijerG instead.

 • The modified Meijer G function is defined by the inverse Laplace transform:

$\mathrm{ModifiedMeijerG}\left(\mathrm{as},\mathrm{bs},\mathrm{cs},\mathrm{ds},z\right)=\frac{1}{2\mathrm{\pi }I}\underset{L}{\oint }\frac{\Gamma \left(1-\mathrm{as}+y\right)\Gamma \left(\mathrm{cs}-y\right)}{\Gamma \left(\mathrm{bs}-y\right)\Gamma \left(1-\mathrm{ds}+y\right)}{ⅇ}^{yz}ⅆy$

 where

$\mathrm{as}=\left[\mathrm{a1},\mathrm{...},\mathrm{am}\right],\mathrm{\Gamma }\left(1-\mathrm{as}+y\right)=\mathrm{\Gamma }\left(1-\mathrm{a1}+y\right)\mathrm{...}\mathrm{\Gamma }\left(1-\mathrm{am}+y\right)$

$\mathrm{bs}=\left[\mathrm{b1},\mathrm{...},\mathrm{bn}\right],\mathrm{\Gamma }\left(\mathrm{bs}-y\right)=\mathrm{\Gamma }\left(\mathrm{b1}-y\right)\mathrm{...}\mathrm{\Gamma }\left(\mathrm{bn}-y\right)$

$\mathrm{cs}=\left[\mathrm{c1},\mathrm{...},\mathrm{cp}\right],\mathrm{\Gamma }\left(\mathrm{cs}-y\right)=\mathrm{\Gamma }\left(\mathrm{c1}-y\right)\mathrm{...}\mathrm{\Gamma }\left(\mathrm{cp}-y\right)$

$\mathrm{ds}=\left[\mathrm{d1},\mathrm{...},\mathrm{dq}\right],\mathrm{\Gamma }\left(1-\mathrm{ds}+y\right)=\mathrm{\Gamma }\left(1-\mathrm{d1}+y\right)\mathrm{...}\mathrm{\Gamma }\left(1-\mathrm{dq}+y\right)$

 and  L is one of three types of integration paths ${L}_{\mathrm{\gamma }+\mathrm{\infty }I}$, ${L}_{\mathrm{\infty }}$, and ${L}_{-\mathrm{\infty }}$.
 Contour ${L}_{\mathrm{\infty }}$ starts at $\mathrm{\infty }+I\mathrm{φ1}$ and finishes at $\mathrm{\infty }+I\mathrm{φ2}$ ($\mathrm{φ1}<\mathrm{φ2}$).
 Contour ${L}_{-\mathrm{\infty }}$ starts at $-\mathrm{\infty }+I\mathrm{φ1}$ and finishes at $-\mathrm{\infty }+I\mathrm{φ2}$ ($\mathrm{φ1}<\mathrm{φ2}$).
 Contour ${L}_{\mathrm{\gamma }+\mathrm{\infty }I}$ starts at $\mathrm{\gamma }+-\mathrm{\infty }$ and finishes at $\mathrm{\gamma }+\mathrm{\infty }I$.
 All the paths ${L}_{\mathrm{\infty }}$, ${L}_{-\mathrm{\infty }}$, and ${L}_{\mathrm{\gamma }+\mathrm{\infty }I}$ put all $\mathrm{cj}+k$ poles on the right and all other poles of the integrand (which must be of the form $\mathrm{aj}-1+k$) on the left.
 • The classical definition of the Meijer G function is related to the modified definition by

${G}_{\mathrm{pq}}^{\mathrm{mn}}\left(z|{}_{{b}_{1},\dots ,{b}_{m},{b}_{m+1},\dots ,{b}_{q}}^{{a}_{1},\dots ,{a}_{n},{a}_{n+1},\dots ,{a}_{p}}\right)=\mathrm{ModifiedMeijerG}\left(\left[{a}_{1},\dots ,{a}_{n}\right],\left[{a}_{n+1},\dots ,{a}_{p}\right],\left[{b}_{1},\dots ,{b}_{m}\right],\left[{b}_{m+1},\dots ,{b}_{q}\right],\mathrm{log}\left(z\right)\right)$

 Note: See Prudnikov, Brychkov, and Marichev.
 • Three noticeable differences between the notations are:
 1 the parameters of the modified Meijer G function are separated out into four natural groups,
 2 ${ⅇ}^{yz}$ instead of ${z}^{y}$ is placed inside the integral definition of ModifiedMeijerG, and
 3 the pq\mn subscripts and superscripts which are now redundant are omitted.

Examples

Important: The ModifiedMeijerG command has been deprecated.  Use the superseding command MeijerG instead.

 > $\mathrm{ModifiedMeijerG}\left(\left[1,1,1\right],\left[1,1\right],\left[1,2\right],\left[2,3,4\right],\mathrm{\pi }\right)$
 ${\mathrm{ModifiedMeijerG}}{}\left(\left[{1}{,}{1}{,}{1}\right]{,}\left[{1}\right]{,}\left[{2}\right]{,}\left[{2}{,}{3}{,}{4}\right]{,}{\mathrm{\pi }}\right)$ (1)
 > $\mathrm{evalf}\left(\right)$
 ${-1.205734962}{}{{10}}^{{-20}}{+}{-0.}{}{I}$ (2)
 > $s≔2\left(\mathrm{sum}\left({\left(-1\right)}^{i}\mathrm{ModifiedMeijerG}\left(\left[\right],\left[\right],\left[0\right],\left[\right],\mathrm{ln}\left(z\right)+\mathrm{ln}\left(1+2I\right)\right),i=0..\mathrm{\infty }\right)\right)$
 ${s}{≔}{2}{}\left({\sum }_{{i}{=}{0}}^{{\mathrm{\infty }}}{}{\left({-1}\right)}^{{i}}{}{\mathrm{ModifiedMeijerG}}{}\left(\left[\right]{,}\left[\right]{,}\left[{0}\right]{,}\left[\right]{,}{\mathrm{ln}}{}\left({z}\right){+}{\mathrm{ln}}{}\left({1}{+}{2}{}{I}\right)\right)\right)$ (3)
 > $\mathrm{convert}\left(s,'\mathrm{StandardFunctions}'\right)$
 ${2}{}\left({\sum }_{{i}{=}{0}}^{{\mathrm{\infty }}}{}{\left({-1}\right)}^{{i}}{}{{ⅇ}}^{\left({-1}{-}{2}{}{I}\right){}{z}}\right)$ (4)
 > $\mathrm{convert}\left(\mathrm{exp}\left(z\right),'\mathrm{ModifiedMeijerG}',z\right)$
 ${\mathrm{ModifiedMeijerG}}{}\left(\left[\right]{,}\left[\right]{,}\left[{0}\right]{,}\left[\right]{,}{\mathrm{ln}}{}\left({z}\right){+}{I}{}{\mathrm{\pi }}\right)$ (5)
 > $\mathrm{convert}\left(\mathrm{sin}\left(z\right),'\mathrm{ModifiedMeijerG}',z\right)$
 $\sqrt{{\mathrm{\pi }}}{}{\mathrm{ModifiedMeijerG}}{}\left(\left[\right]{,}\left[\right]{,}\left[\frac{{1}}{{2}}\right]{,}\left[{0}\right]{,}{2}{}{\mathrm{ln}}{}\left({z}\right){-}{2}{}{\mathrm{ln}}{}\left({2}\right)\right)$ (6)
 > $\mathrm{convert}\left(\mathrm{cos}\left(z\right),'\mathrm{ModifiedMeijerG}',z\right)$
 $\sqrt{{\mathrm{\pi }}}{}{\mathrm{ModifiedMeijerG}}{}\left(\left[\right]{,}\left[\right]{,}\left[{0}\right]{,}\left[\frac{{1}}{{2}}\right]{,}{2}{}{\mathrm{ln}}{}\left({z}\right){-}{2}{}{\mathrm{ln}}{}\left({2}\right)\right)$ (7)
 > $\mathrm{convert}\left(\mathrm{Ei}\left(z\right),'\mathrm{ModifiedMeijerG}',z\right)$
 ${-}{\mathrm{ModifiedMeijerG}}{}\left(\left[\right]{,}\left[{1}\right]{,}\left[{0}{,}{0}\right]{,}\left[\right]{,}{\mathrm{ln}}{}\left({z}\right){+}{I}{}{\mathrm{\pi }}\right)$ (8)

References

 Prudnikov, A. P.; Brychkov, Yu; and Marichev, O. Integrals and Series, Volume 3: More Special Functions. Gordon and Breach Science, 1990.

 See Also