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Erfc






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfc[z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself





http://functions.wolfram.com/06.27.06.0016.01









  


  










Input Form





Erfc[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == 1 - ((2 z)/Sqrt[Pi]) Sum[((-1)^k z^(2 k))/((2 k + 1) k!), {k, 0, n}] == Erfc[z] - (((-1)^n 2 z^(3 + 2 n))/(Sqrt[Pi] (3 + 2 n) (n + 1)!)) HypergeometricPFQ[{1, n + 3/2}, {n + 2, n + 5/2}, -z^2] && Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erfc", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], SqrtBox["\[Pi]"]]]], "\[Equal]", RowBox[List[RowBox[List["Erfc", "[", "z", "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "2", " ", SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["n", "+", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["5", "2"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02