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ModifiedMeijerG

modified Meijer G function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ModifiedMeijerG(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

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:

ModifiedMeijerGas,bs,cs,ds,z=12πILΓ1as+yΓcsyΓbsyΓ1ds+yⅇyzⅆy

  

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 definition of the Meijer G function is related to the modified definition by

Gpqmn(z|b1,,bm,bm+1,,bqa1,,an,an+1,,ap)=ModifiedMeijerGa1,,an,an+1,,ap,b1,,bm,bm+1,,bq,logz

  

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. 

&ExponentialE;yz instead of zy 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.

ModifiedMeijerG1&comma;1&comma;1&comma;1&comma;1&comma;1&comma;2&comma;2&comma;3&comma;4&comma;π

ModifiedMeijerG1&comma;1&comma;1&comma;1&comma;2&comma;2&comma;3&comma;4&comma;π

(1)

evalf

−1.20573496210−20+−0.I

(2)

s2sum1iModifiedMeijerG&comma;&comma;0&comma;&comma;lnz+ln1+2I&comma;i=0..

s2i=0−1iModifiedMeijerG&comma;&comma;0&comma;&comma;lnz+ln1+2I

(3)

converts&comma;StandardFunctions

2i=0−1i&ExponentialE;−12Iz

(4)

convertexpz&comma;ModifiedMeijerG&comma;z

ModifiedMeijerG&comma;&comma;0&comma;&comma;lnz+Iπ

(5)

convertsinz&comma;ModifiedMeijerG&comma;z

πModifiedMeijerG&comma;&comma;12&comma;0&comma;2lnz2ln2

(6)

convertcosz&comma;ModifiedMeijerG&comma;z

πModifiedMeijerG&comma;&comma;0&comma;12&comma;2lnz2ln2

(7)

convertEiz&comma;ModifiedMeijerG&comma;z

ModifiedMeijerG&comma;1&comma;0&comma;0&comma;&comma;lnz+Iπ

(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

convert/MeijerG

convert/StandardFunctions

MeijerG