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Erfc




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfc[z] > Integral representations > Contour integral representations





http://functions.wolfram.com/06.27.07.0005.01









  


  








Input Form





Erfc[z] == (1/(Sqrt[Pi] 2 Pi I)) Integrate[(Gamma[s] Gamma[s + 1/2])/Gamma[s + 1]/z^(2 s), {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; 0 < \[Gamma] && Abs[Arg[z]] < Pi/2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Erfc", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["1", " "]], RowBox[List[SqrtBox["\[Pi]"], "2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erfc", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]]], RowBox[List["\[DifferentialD]", "s"]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", "2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]]]]]


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





2001-10-29