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MeijerG

Meijer G function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MeijerG([as, bs], [cs, ds], z)

Parameters

as

-

list of the form [a1, ..., am]; first group of numerator Γ parameters

bs

-

list of the form [b1, ..., bn]; first group of denominator Γ parameters

cs

-

list of the form [c1, ..., cp]; second group of numerator Γ parameters

ds

-

list of the form [d1, ..., dq]; second group of denominator Γ parameters

z

-

expression

Description

• 

The Meijer G function is defined by the inverse Laplace transform

MeijerGas,bs,cs,ds,z=12πILΓ1as+yΓcsyΓbsyΓ1ds+yzyⅆy

  

where

as=a1,...,am,Γ1as+y=Γ1a1+y...Γ1am+y

bs=b1,...,bn,Γbsy=Γb1y...Γbny

cs=c1,...,cp,Γcsy=Γc1y...Γcpy

ds=d1,...,dq,Γ1ds+y=Γ1d1+y...Γ1dq+y

  

and  L is one of three types of integration paths Lγ+I, L, and L.

  

Contour L starts at +I&phi;1 and finishes at +I&phi;2&phi;1<&phi;2.

  

Contour L starts at +I&phi;1 and finishes at +I&phi;2&phi;1<&phi;2.

  

Contour Lγ+I starts at γ+ and finishes at γ+I.

  

All the paths L, L, and Lγ+I put all cj+k poles on the right and all other poles of the integrand (which must be of the form aj1+k) on the left.

• 

The classical notation used to represent the MeijerG function relates to the notation used in Maple by

Gpqmn(z|b1,,bm,bm+1,,bqa1,,an,an+1,,ap)=MeijerGa1,,an,an+1,,ap,b1,,bm,bm+1,,bq,z

  

Note: See Prudnikov, Brychkov, and Marichev.

  

The MeijerG function satisfies the following qth-order linear differential equation

−1pmnxi=1pxDai+1i=1qxDbiyx=0

  

where D=ddx and p is less than or equal to q.

Examples

MeijerG1&comma;1&comma;1&comma;1&comma;&comma;&comma;4&comma;3&comma;2&comma;2&comma;Pi

MeijerG1&comma;1&comma;1&comma;1&comma;&comma;&comma;4&comma;3&comma;2&comma;2&comma;π

(1)

evalf

8.89830817810−28+9.79667712510−26I

(2)

sMeijerG&comma;&comma;0&comma;&comma;z1&plus;2I

sMeijerG&comma;&comma;0&comma;&comma;1+2Iz

(3)

converts&comma;&apos;StandardFunctions&apos;

&ExponentialE;−12Iz

(4)

convert&ExponentialE;z&comma;&apos;MeijerG&apos;&comma;include&equals;elementary

MeijerG&comma;&comma;0&comma;&comma;z

(5)

convertsinz&comma;&apos;MeijerG&apos;&comma;include&equals;elementary

πMeijerG&comma;&comma;12&comma;0&comma;z24

(6)

convertcosz&comma;&apos;MeijerG&apos;&comma;include&equals;elementary

πMeijerG&comma;&comma;0&comma;12&comma;z24

(7)

convertEiz&comma;&apos;MeijerG&apos;

MeijerG&comma;1&comma;0&comma;0&comma;&comma;z

(8)

References

  

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

See Also

Appell

convert/StandardFunctions

dpolyform

Heun

hypergeom

hyperode