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MeijerG - Meijer G function
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Calling Sequence
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MeijerG([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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The Meijer G function is defined by the inverse Laplace transform
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![MeijerG([as,bs],[cs,ds],z)=1/(2 Pi I) (∮)(GAMMA(1-as+y) GAMMA(cs-y))/(GAMMA(bs-y) GAMMA(1-ds+y)) z^ydy](/support/helpjp/helpview.aspx?si=5465/file00102/math104.png)
![as = [a1, `...`, am], GAMMA(1-as+y) = GAMMA(1-a1+y)*`...`*GAMMA(1-am+y)](/support/helpjp/helpview.aspx?si=5465/file00102/math111.png)
![bs = [b1, `...`, bn], GAMMA(bs-y) = GAMMA(b1-y)*`...`*GAMMA(bn-y)](/support/helpjp/helpview.aspx?si=5465/file00102/math116.png)
![cs = [c1, `...`, cp], GAMMA(cs-y) = GAMMA(c1-y)*`...`*GAMMA(cp-y)](/support/helpjp/helpview.aspx?si=5465/file00102/math121.png)
![ds = [d1, `...`, dq], GAMMA(1-ds+y) = GAMMA(1-d1+y)*`...`*GAMMA(1-dq+y)](/support/helpjp/helpview.aspx?si=5465/file00102/math126.png)
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The classical notation used to represent the MeijerG function relates to the notation used in Maple 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]))=MeijerG([[a[1], .., a[n]],[a[n+1], .., a[p]]],[[b[1], .., b[m]],[b[m+1], .., b[q]]],z)](/support/helpjp/helpview.aspx?si=5465/file00102/math181.png)
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Note: See Prudnikov, Brychkov, and Marichev.
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The MeijerG function satisfies the following  th-order linear differential equation
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![((-1)^(p-m-n)*x*(product(x*D-a[i]+1, i = 1 .. p))-product(x*D-b[i], i = 1 .. q))*y(x) = 0](/support/helpjp/helpview.aspx?si=5465/file00102/math195.png)
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where  and p is less than or equal to q.
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Examples
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![MeijerG([[1, 1, 1, 1], []], [[], [4, 3, 2, 2]], Pi)](/support/helpjp/helpview.aspx?si=5465/file00102/math211.png)
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![MeijerG([[1, 1, 1, 1], []], [[], [4, 3, 2, 2]], Pi)](/support/helpjp/helpview.aspx?si=5465/file00102/math214.png)
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![evalf(MeijerG([[1, 1, 1, 1], []], [[], [4, 3, 2, 2]], Pi))](/support/helpjp/helpview.aspx?si=5465/file00102/math218.png)
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| (2) |
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![s := MeijerG([[], []], [[0], []], z*(1+2*I))](/support/helpjp/helpview.aspx?si=5465/file00102/math225.png)
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![s := MeijerG([[], []], [[0], []], (1+2*I)*z)](/support/helpjp/helpview.aspx?si=5465/file00102/math228.png)
| (3) |
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| (4) |
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![MeijerG([[], []], [[0], []], -z)](/support/helpjp/helpview.aspx?si=5465/file00102/math242.png)
| (5) |
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![Pi^(1/2)*MeijerG([[], []], [[1/2], [0]], (1/4)*z^2)](/support/helpjp/helpview.aspx?si=5465/file00102/math249.png)
| (6) |
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![Pi^(1/2)*MeijerG([[], []], [[0], [1/2]], (1/4)*z^2)](/support/helpjp/helpview.aspx?si=5465/file00102/math256.png)
| (7) |
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![-MeijerG([[], [1]], [[0, 0], []], -z)](/support/helpjp/helpview.aspx?si=5465/file00102/math263.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. New York: Gordon and Breach Science Publishers, 1990.
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