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ModifiedMeijerG - modified Meijer G function
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Calling Sequence
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ModifiedMeijerG(as, bs, cs, ds, z)
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Parameters
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as
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list of the form [a1, ..., am]; first group of numerator  parameters
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bs
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list of the form [b1, ..., bn]; first group of denominator  parameters
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cs
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list of the form [c1, ..., cp]; second group of numerator  parameters
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ds
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list of the form [d1, ..., dq]; second group of denominator  parameters
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z
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expression
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Description
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•
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The modified Meijer G function is defined by the inverse Laplace transform:
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![as = [a1, `...`, am], GAMMA(1-as+y) = GAMMA(1-a1+y)*`...`*GAMMA(1-am+y)](/support/helpjp/helpview.aspx?si=6313/file00104/math111.png)
![bs = [b1, `...`, bn], GAMMA(bs-y) = GAMMA(b1-y)*`...`*GAMMA(bn-y)](/support/helpjp/helpview.aspx?si=6313/file00104/math116.png)
![cs = [c1, `...`, cp], GAMMA(cs-y) = GAMMA(c1-y)*`...`*GAMMA(cp-y)](/support/helpjp/helpview.aspx?si=6313/file00104/math121.png)
![ds = [d1, `...`, dq], GAMMA(1-ds+y) = GAMMA(1-d1+y)*`...`*GAMMA(1-dq+y)](/support/helpjp/helpview.aspx?si=6313/file00104/math126.png)
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The classical definition of the Meijer G function is related to the modified definition by
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![G[pq]^mn(z|( )[b[1], .., b[m],b[m+1], .., b[q]]^(a[1], .., a[n],a[n+1], .., a[p]))=ModifiedMeijerG([a[1], .., a[n]],[a[n+1], .., a[p]],[b[1], .., b[m]],[b[m+1], .., b[q]],log(z))](/support/helpjp/helpview.aspx?si=6313/file00104/math183.png)
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Note: See Prudnikov, Brychkov, and Marichev.
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Three noticeable differences between the notations are:
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1.
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the parameters of the modified Meijer G function are separated out into four natural groups,
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2.
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 instead of  is placed inside the integral definition of ModifiedMeijerG, and
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3.
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the pq\mn subscripts and superscripts which are now redundant are omitted.
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Use of this function in Maple is not recommended. MeijerG should be used instead. This function is provided only for backward compatibility.
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Examples
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>
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![ModifiedMeijerG([1, 1, 1], [1, 1], [1, 2], [2, 3, 4], Pi)](/support/helpjp/helpview.aspx?si=6313/file00104/math213.png)
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![ModifiedMeijerG([1, 1, 1], [1], [2], [2, 3, 4], Pi)](/support/helpjp/helpview.aspx?si=6313/file00104/math216.png)
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![evalf(ModifiedMeijerG([1, 1, 1], [1], [2], [2, 3, 4], Pi))](/support/helpjp/helpview.aspx?si=6313/file00104/math220.png)
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| (2) |
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![s := 2*(sum((-1)^i*ModifiedMeijerG([], [], [0], [], ln(z)+ln(1+2*I)), i = 0 .. infinity))](/support/helpjp/helpview.aspx?si=6313/file00104/math227.png)
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![s := 2*(sum((-1)^i*ModifiedMeijerG([], [], [0], [], ln(z)+ln(1+2*I)), i = 0 .. infinity))](/support/helpjp/helpview.aspx?si=6313/file00104/math230.png)
| (3) |
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>
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| (4) |
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![ModifiedMeijerG([], [], [0], [], ln(z)+I*Pi)](/support/helpjp/helpview.aspx?si=6313/file00104/math244.png)
| (5) |
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![Pi^(1/2)*ModifiedMeijerG([], [], [1/2], [0], 2*ln(z)-2*ln(2))](/support/helpjp/helpview.aspx?si=6313/file00104/math251.png)
| (6) |
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![Pi^(1/2)*ModifiedMeijerG([], [], [0], [1/2], 2*ln(z)-2*ln(2))](/support/helpjp/helpview.aspx?si=6313/file00104/math258.png)
| (7) |
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![-ModifiedMeijerG([], [1], [0, 0], [], ln(z)+I*Pi)](/support/helpjp/helpview.aspx?si=6313/file00104/math265.png)
| (8) |
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References
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Prudnikov, A. P.; Brychkov, Yu; and Marichev, O. Integrals and Series, Volume 3: More Special Functions. Gordon and Breach Science, 1990.
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