html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 MeijerG

 http://functions.wolfram.com/07.34.06.0023.01

 Input Form

 MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, q + 2]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z] \[Proportional] Sum[((Product[If[j == k, 1, Gamma[Subscript[b, j] - Subscript[b, k]]], {j, 1, m}] Product[Gamma[1 + Subscript[b, k] - Subscript[a, j]], {j, 1, n}])/(Product[Gamma[Subscript[a, j] - Subscript[b, k]], {j, n + 1, q + 2}] Product[Gamma[1 - Subscript[b, j] + Subscript[b, k]], {j, m + 1, q}])) z^Subscript[b, k] (1 + O[z]), {k, 1, m}] + (Pi^(m + n - q - 3/2)/2) Sum[(Product[Sin[Pi (Subscript[a, r] - Subscript[b, j])], {j, m + 1, q}]/ Product[If[j == r, 1, Sin[Pi (Subscript[a, r] - Subscript[a, j])]], {j, 1, n}]) z^(Subscript[a, r] - 1) ((-1)^(q - m - n - 1)/z)^ (\[Chi] + Subscript[a, r] - 1) (Exp[I (Pi (\[Chi] + Subscript[a, r] - 1) + 2 Sqrt[(-1)^(q - m - n - 1)/z])] (1 + O[Sqrt[z]]) + Exp[(-I) (Pi (\[Chi] + Subscript[a, r] - 1) + 2 Sqrt[(-1)^(q - m - n - 1)/z])] (1 + O[Sqrt[z]])), {r, 1, n}] /; (z -> 0) && \[Chi] == (1/2) (Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, q + 2}] + 3/2) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= m && 1 <= k <= m, !Element[Subscript[b, j] - Subscript[b, k], Integers]]

 Standard Form

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RowBox[List[FractionBox[SuperscriptBox["\[Pi]", RowBox[List["m", "+", "n", "-", "q", "-", FractionBox["3", "2"]]]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["m", "+", "1"]]]], "q"], RowBox[List["Sin", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[SubscriptBox["a", "r"], "-", SubscriptBox["b", "j"]]], ")"]]]], "]"]], " "]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "r"]], ",", "1", ",", RowBox[List["Sin", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[SubscriptBox["a", "r"], "-", SubscriptBox["a", "j"]]], ")"]]]], "]"]]]], "]"]]]]], SuperscriptBox["z", RowBox[List[SubscriptBox["a", "r"], "-", "1"]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", 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"m"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], ")"]]]], ")"]]]]]]]]]]

 MathML Form

 G q + 2 , q m , n ( z a 1 , , a n , a n + 1 , , a q + 2 b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["q", "+", "2"]], ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", 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", RowBox[List["q", "+", "2"]]], 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, True]] k = 1 m j = 1 j k m Γ ( b j - b k ) j = 1 n Γ ( 1 - a j + b k ) j = n + 1 q + 2 Γ ( a j - b k ) j = m + 1 q Γ ( 1 - b j + b k ) z b k ( 1 + O ( z ) ) + π m + n - q - 3 2 2 r = 1 n j = m + 1 q sin ( π ( a r - b j ) ) j = 1 j r n sin ( π ( a r - a j ) ) z a r - 1 ( ( - 1 ) q - m - n - 1 z ) χ + a r - 1 ( exp ( ( π ( χ + a r - 1 ) + 2 ( - 1 ) q - m - n - 1 z ) ) ( 1 + O ( z ) ) + exp ( - ( π ( χ + a r - 1 ) + 2 ( - 1 ) q - m - n - 1 z ) ) ( 1 + O ( z ) ) ) /; ( z "\[Rule]" 0 ) χ 1 2 ( j = 1 q b j - j = 1 q + 2 a j + 3 2 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j m 1 k m ( b j - b k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) G q + 2 , q m , n ( z a 1 , , a n , a n + 1 , , a q + 2 b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["q", "+", "2"]], ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", 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", RowBox[List["q", "+", "2"]]], 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, True]] k = 1 m j = 1 j k m Γ ( b j - b k ) j = 1 n Γ ( 1 - a j + b k ) j = n + 1 q + 2 Γ ( a j - b k ) j = m + 1 q Γ ( 1 - b j + b k ) z b k ( 1 + O ( z ) ) + π m + n - q - 3 2 2 r = 1 n j = m + 1 q sin ( π ( a r - b j ) ) j = 1 j r n sin ( π ( a r - a j ) ) z a r - 1 ( ( - 1 ) q - m - n - 1 z ) χ + a r - 1 ( exp ( ( π ( χ + a r - 1 ) + 2 ( - 1 ) q - m - n - 1 z ) ) ( 1 + O ( z ) ) + exp ( - ( π ( χ + a r - 1 ) + 2 ( - 1 ) q - m - n - 1 z ) ) ( 1 + O ( z ) ) ) /; ( z "\[Rule]" 0 ) χ 1 2 ( j = 1 q b j - j = 1 q + 2 a j + 3 2 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j m 1 k m ( b j - b k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2001-10-29