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FresnelC






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.33.06.0017.01









  


  










Input Form





FresnelC[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == z Sum[(Pi^(2 k) (-z^4)^k)/(2^(2 k) ((4 k + 1) (2 k)!)), {k, 0, n}] == FresnelC[z] + (((-1)^n 4^(-1 - n) Pi^(2 (1 + n)) z^(5 + 4 n))/ ((5 + 4 n) (2 n + 2)!)) HypergeometricPFQ[{1, 5/4 + n}, {3/2 + n, 2 + n, 9/4 + n}, (-(1/16)) Pi^2 z^4] && 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["FresnelC", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "4"]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["FresnelC", "[", "z", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["4", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List["5", "+", RowBox[List["4", " ", "n"]]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["5", "4"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["9", "4"], "+", "n"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "4"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["5", "+", RowBox[List["4", " ", "n"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02