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/07.34.03.0237.01

 Input Form

 MeijerG[{{}, {}}, {{b}, {b + 1/2 - n}}, z] == ((-(I/Sqrt[Pi])) z^b (I Cos[(n Pi)/2 - 2 I Sqrt[-z]] Sum[Gamma[1/2 + 2 k + Abs[-(1/2) + n]]/(16^k (-z)^k ((2 k)! Gamma[1/2 - 2 k + Abs[-(1/2) + n]])), {k, 0, Floor[(1/4) (-1 + 2 Abs[-(1/2) + n])]}] + (1/Sqrt[-z]) (Sin[(n Pi)/2 - 2 I Sqrt[-z]] Sum[(2^(-2 - 4 k) (1/2 + 2 k + Abs[-(1/2) + n])!)/ ((-z)^k ((-(3/2) - 2 k + Abs[-(1/2) + n])! Gamma[2 + 2 k])), {k, 0, Floor[(1/4) (-3 + 2 Abs[-(1/2) + n])]}])))/ (E^((1/2) I n Pi) (-z)^(n/2)) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List["b", "+", FractionBox["1", "2"], "-", "n"]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", SqrtBox["\[Pi]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "n", " ", "\[Pi]"]]], " ", SuperscriptBox["z", "b"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "n"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]]]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox["16", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], "]"]]]]]]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]]]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["4", " ", "k"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]], "]"]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 G 0 , 2 1 , 0 ( z b , b + 1 2 - n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "2"]], RowBox[List["1", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["1", "2"], "-", "n"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] - π - π n 2 z b ( - z ) - n 2 ( 1 - z sin ( n π 2 - 2 - z ) k = 0 1 4 ( 2 "\[LeftBracketingBar]" n - 1 2 "\[RightBracketingBar]" - 3 ) 2 - 4 k - 2 ( - z ) - k ( 2 k + "\[LeftBracketingBar]" n - 1 2 "\[RightBracketingBar]" + 1 2 ) ! ( "\[LeftBracketingBar]" n - 1 2 "\[RightBracketingBar]" - 3 2 - 2 k ) ! Γ ( 2 k + 2 ) + cos ( n π 2 - 2 - z ) k = 0 1 4 ( 2 "\[LeftBracketingBar]" n - 1 2 "\[RightBracketingBar]" - 1 ) 16 - k ( - z ) - k Γ ( 2 k + "\[LeftBracketingBar]" n - 1 2 "\[RightBracketingBar]" + 1 2 ) ( 2 k ) ! Γ ( "\[LeftBracketingBar]" n - 1 2 "\[RightBracketingBar]" + 1 2 - 2 k ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition MeijerG b b 1 2 -1 n z -1 1 2 -1 -1 n 2 -1 z b -1 z -1 n 2 -1 1 -1 z 1 2 -1 n 2 -1 -1 2 -1 z 1 2 k 0 1 4 2 n -1 1 2 -3 2 -4 k -2 -1 z -1 k 2 k n -1 1 2 1 2 n -1 1 2 -1 3 2 -1 2 k Gamma 2 k 2 -1 n 2 -1 -1 2 -1 z 1 2 k 0 1 4 2 n -1 1 2 -1 16 -1 k -1 z -1 k Gamma 2 k n -1 1 2 1 2 2 k Gamma n -1 1 2 1 2 -1 2 k -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "b_", "}"]], ",", RowBox[List["{", RowBox[List["b_", "+", FractionBox["1", "2"], "-", "n_"]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "n", " ", "\[Pi]"]]], " ", SuperscriptBox["z", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", FractionBox["n", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]]]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox["16", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], "]"]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]]]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["4", " ", "k"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], "]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]], "]"]]]]]]]]], SqrtBox[RowBox[List["-", "z"]]]]]], ")"]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29