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/07.34.03.0053.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", RowBox[List["a", "-", FractionBox["1", "2"], "+", "n"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", SqrtBox["\[Pi]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "n", " ", "\[Pi]"]]], " ", SuperscriptBox["z", RowBox[List["a", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], 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", " ", RowBox[List["\[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"]], ")"]], "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[SqrtBox[RowBox[List["-", "z"]]], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["\[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"]], ")"]], "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 2 , 0 0 , 1 ( z a , a + n - 1 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "0"]], RowBox[List["0", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n", "-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] - π - π n 2 z a - 1 ( - z ) n / 2 ( - 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 a a n -1 1 2 z -1 1 2 -1 -1 n 2 -1 z a -1 -1 z n 2 -1 -1 z 1 2 n 2 -1 -1 2 -1 z 1 2 -1 k 0 1 4 2 n -1 1 2 -3 2 -4 k -2 -1 z 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 -1 k 0 1 4 2 n -1 1 2 -1 16 -1 k -1 z 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

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

 2001-10-29