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/07.34.06.0044.01

 Input Form

 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]}}, z] \[Proportional] Sum[((Product[If[j == k, 1, Gamma[Subscript[a, k] - Subscript[a, j]]], {j, 1, n}] Product[Gamma[1 + Subscript[b, j] - Subscript[a, k]], {j, 1, m}])/(Product[Gamma[Subscript[a, k] - Subscript[b, j]], {j, m + 1, q}] Product[Gamma[1 + Subscript[a, j] - Subscript[a, k]], {j, n + 1, p}])) z^(Subscript[a, k] - 1) (1 + O[1/z]), {k, 1, n}] + KroneckerDelta[q, p + 1] Subscript[d, 1] + KroneckerDelta[q, p + 2] Subscript[d, 2] + (UnitStep[q - p - 2] - KroneckerDelta[q, p + 2] - KroneckerDelta[q, p + 1]) (1 - KroneckerDelta[q, p + 1]) Subscript[d, 3] /; (Abs[z] -> Infinity) && \[Beta] == q - p && Subscript[d, 1] == Pi^(m + n - p - 1) Exp[(-1)^(p - m - n) z] Sum[(Product[Sin[Pi (Subscript[a, j] - Subscript[b, r])], {j, n + 1, p}]/ Product[If[j == r, 1, Sin[Pi (Subscript[b, j] - Subscript[b, r])]], {j, 1, m}]) z^Subscript[b, r] ((-1)^(p - m - n) z)^ (\[Chi] - Subscript[b, r]) (1 + O[1/z]), {r, 1, m}] && Subscript[d, 2] == Pi^(m + n - p - 3/2) Sum[(Product[Sin[Pi (Subscript[a, j] - Subscript[b, r])], {j, n + 1, p}]/ Product[If[j == r, 1, Sin[Pi (Subscript[b, j] - Subscript[b, r])]], {j, 1, m}]) z^Subscript[b, r] ((-1)^(p - m - n - 1) z)^ (\[Chi] - Subscript[b, r]) Cos[2 Sqrt[(-1)^(p - m - n - 1) z] + Pi (\[Chi] - Subscript[b, r])] (1 + O[1/Sqrt[z]]), {r, 1, m}] && Subscript[d, 3] == ((2 (2 Pi)^((1 - \[Beta])/2) Pi^(m + n - p - 1))/ Sqrt[\[Beta]]) Exp[\[Beta] Cos[(Pi (p - m - n))/\[Beta]] z^(1/\[Beta])] Sum[(Product[Sin[Pi (Subscript[a, j] - Subscript[b, r])], {j, n + 1, p}]/ Product[If[j == r, 1, Sin[Pi (Subscript[b, j] - Subscript[b, r])]], {j, 1, m}]) Cos[Pi (p - m - n) (\[Chi] - Subscript[b, r]) + \[Beta] Sin[(Pi (p - m - n))/\[Beta]] z^(1/\[Beta])] (1 + O[1/z^(1/\[Beta])]), {r, 1, m}] && \[Chi] == (1/\[Beta]) (Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, p}] + (1 - \[Beta])/2) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

 Standard Form

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 MathML Form

 G p , q m , n ( z a 1 , , a n , a n + 1 , , a p b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "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", "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, True]] k = 1 n j = 1 j k n Γ ( a k - a j ) j = 1 m Γ ( 1 - a k + b j ) j = m + 1 q Γ ( a k - b j ) j = n + 1 p Γ ( a j - a k + 1 ) z a k - 1 ( 1 + O ( 1 z ) ) + δ KroneckerDelta q , p + 1 d 1 + δ KroneckerDelta q , p + 2 d 2 + ( 1 - δ KroneckerDelta q , p + 1 ) ( θ UnitStep ( - p + q - 2 ) - δ KroneckerDelta q , p + 1 - δ KroneckerDelta q , p + 2 ) d 3 /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) β q - p d 1 π m + n - p - 1 exp ( ( - 1 ) p - m - n z ) r = 1 m j = n + 1 p sin ( π ( a j - b r ) ) j = 1 j r m sin ( π ( b j - b r ) ) z b r ( ( - 1 ) p - m - n z ) χ - b r ( 1 + O ( 1 z ) ) d 2 π m + n - p - 3 2 r = 1 m j = n + 1 p sin ( π ( a j - b r ) ) j = 1 j r m sin ( π ( b j - b r ) ) z b r ( ( - 1 ) p - m - n - 1 z ) χ - b r cos ( π ( χ - b r ) + 2 ( - 1 ) p - m - n - 1 z ) ( 1 + O ( 1 z ) ) d 3 2 ( 2 π ) 1 - β 2 π m + n - p - 1 β exp ( β cos ( π ( p - m - n ) β ) z 1 / β ) r = 1 m j = n + 1 p sin ( π ( a j - b r ) ) j = 1 j r m sin ( π ( b j - b r ) ) cos ( π ( p - m - n ) ( χ - b r ) + β sin ( π ( p - m - n ) β ) z 1 / β ) ( 1 + O ( 1 z 1 / β ) ) χ 1 β ( 1 - β 2 - j = 1 p a j + j = 1 q b j ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j n 1 k n ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) G p , q m , n ( z a 1 , , a n , a n + 1 , , a p b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "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", "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, True]] k = 1 n j = 1 j k n Γ ( a k - a j ) j = 1 m Γ ( 1 - a k + b j ) j = m + 1 q Γ ( a k - b j ) j = n + 1 p Γ ( a j - a k + 1 ) z a k - 1 ( 1 + O ( 1 z ) ) + δ KroneckerDelta q , p + 1 d 1 + δ KroneckerDelta q , p + 2 d 2 + ( 1 - δ KroneckerDelta q , p + 1 ) ( θ UnitStep ( - p + q - 2 ) - δ KroneckerDelta q , p + 1 - δ KroneckerDelta q , p + 2 ) d 3 /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) β q - p d 1 π m + n - p - 1 exp ( ( - 1 ) p - m - n z ) r = 1 m j = n + 1 p sin ( π ( a j - b r ) ) j = 1 j r m sin ( π ( b j - b r ) ) z b r ( ( - 1 ) p - m - n z ) χ - b r ( 1 + O ( 1 z ) ) d 2 π m + n - p - 3 2 r = 1 m j = n + 1 p sin ( π ( a j - b r ) ) j = 1 j r m sin ( π ( b j - b r ) ) z b r ( ( - 1 ) p - m - n - 1 z ) χ - b r cos ( π ( χ - b r ) + 2 ( - 1 ) p - m - n - 1 z ) ( 1 + O ( 1 z ) ) d 3 2 ( 2 π ) 1 - β 2 π m + n - p - 1 β exp ( β cos ( π ( p - m - n ) β ) z 1 / β ) r = 1 m j = n + 1 p sin ( π ( a j - b r ) ) j = 1 j r m sin ( π ( b j - b r ) ) cos ( π ( p - m - n ) ( χ - b r ) + β sin ( π ( p - m - n ) β ) z 1 / β ) ( 1 + O ( 1 z 1 / β ) ) χ 1 β ( 1 - β 2 - j = 1 p a j + j = 1 q b j ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j n 1 k n ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) [/itex]

 Rule Form

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

 2001-10-29