MeijerG - Maple Help

convert/MeijerG

convert an expression to MeijerG form

 Calling Sequence convert( $\mathrm{expr}$, MeijerG, opt_1, opt_2, $...$ )

Parameters

 expr - expression opt_i - optional arguments, see convert/to_special_function

Description

 • The convert( expr, 'MeijerG') command converts the functions found in expr into their equivalent MeijerG representation, whenever possible. Most special and all elementary functions can be so converted.

Examples

 > $\mathrm{BesselK}\left(n,z\right):$
 > $\mathrm{BesselK}\left(n,z\right)=\mathrm{convert}\left(\mathrm{BesselK}\left(n,z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{BesselK}}{}\left({n}{,}{z}\right){=}\frac{{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[\frac{{n}}{{2}}{,}{-}\frac{{n}}{{2}}\right]{,}\left[\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right)}{{2}}$ (1)
 > $\mathrm{hypergeom}\left(\left[a,b\right],\left[c\right],z\right):$
 > $\mathrm{hypergeom}\left(\left[a,b\right],\left[c\right],z\right)=\mathrm{convert}\left(\mathrm{hypergeom}\left(\left[a,b\right],\left[c\right],z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{a}{,}{b}\right]{,}\left[{c}\right]{,}{z}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}{\mathrm{MeijerG}}{}\left(\left[\left[{1}{-}{a}{,}{1}{-}{b}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{1}{-}{c}\right]\right]{,}{-}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}$ (2)

Elementary functions are not converted by default (see convert/to_special_function)

 > $\mathrm{convert}\left(\mathrm{sin}\left(z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{sin}}{}\left({z}\right)$ (3)

To convert them use either of the optional arguments: include=elementary or include=all

 > $\mathrm{sin}\left(z\right):$
 > $\mathrm{sin}\left(z\right)=\mathrm{convert}\left(\mathrm{sin}\left(z\right),\mathrm{MeijerG},\mathrm{include}=\mathrm{all}\right)$
 ${\mathrm{sin}}{}\left({z}\right){=}\sqrt{{\mathrm{\pi }}}{}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[\frac{{1}}{{2}}\right]{,}\left[{0}\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right)$ (4)

References

 Prudnikov, A.P.; Brychkov, Yu; and Marichev, O. Integrals and Series. Gordon and Breach Science, 1990. Vol. 3: More Special Functions.
 Roach, K. "Meijer G Function Representations." In Proceedings of ISSAC '97, pp. 205-211. Edited by Wolfgang Kuchlin. New York: ACM Press, 1997.