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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Differentiation
Symbolic differentiation
With respect to z
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http://functions.wolfram.com/07.34.20.0017.02
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D[MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]},
{Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}},
{{Subscript[b, 1], \[Ellipsis], Subscript[b, m]},
{Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, w z^(r/l)],
{z, u}] == (l^\[Mu] r^u MeijerG[{{0, 1/r, \[Ellipsis], (r - 1)/r,
Subscript[a, 1]/l, \[Ellipsis], (Subscript[a, 1] + l - 1)/l,
\[Ellipsis], Subscript[a, n]/l, \[Ellipsis], (Subscript[a, n] + l - 1)/
l}, {Subscript[a, n + 1]/l, \[Ellipsis],
(Subscript[a, n + 1] + l - 1)/l, \[Ellipsis], Subscript[a, p]/l,
\[Ellipsis], (Subscript[a, p] + l - 1)/l}},
{{Subscript[b, 1]/l, \[Ellipsis], (Subscript[b, 1] + l - 1)/l,
\[Ellipsis], Subscript[b, m]/l, \[Ellipsis], (Subscript[b, m] + l - 1)/
l}, {u/r, \[Ellipsis], (u + r - 1)/r, Subscript[b, m + 1]/l,
\[Ellipsis], (Subscript[b, m + 1] + l - 1)/l, \[Ellipsis],
Subscript[b, q]/l, \[Ellipsis], (Subscript[b, q] + l - 1)/l}},
w^l l^(l (p - q)) z^r])/((2 Pi)^((l - 1) SuperStar[c]) z^u) /;
Element[u, Integers] && u >= 0 && SuperStar[c] == m + n - (p + q)/2 &&
\[Mu] == Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j],
{j, 1, p}] + (p - q)/2 + 1
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> u </mi> </msup> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> r </mi> <mo> / </mo> <mi> l </mi> </mrow> </msup> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["w", " ", SuperscriptBox["z", RowBox[List["r", "/", "l"]]]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> u </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - 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</mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> - 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Date Added to functions.wolfram.com (modification date)
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| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | | |
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){kind=small}
