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FresnelS




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.32.06.0017.01









  


  








Input Form





FresnelS[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == z^3 Sum[(2^(-2 k - 1) Pi^(2 k + 1) (-z^4)^k)/ ((4 k + 3) (2 k + 1)!), {k, 0, n}] == FresnelS[z] + (((-1)^n 2^(-3 - 2 n) Pi^(3 + 2 n) z^(7 + 4 n))/ ((7 + 4 n) Gamma[4 + 2 n])) HypergeometricPFQ[{1, 7/4 + n}, {2 + n, 5/2 + n, 11/4 + 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["FresnelS", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "4"]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "3"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["FresnelS", "[", "z", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["7", "+", RowBox[List["4", " ", "n"]]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["7", "4"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]], ",", RowBox[List[FractionBox["11", "4"], "+", "n"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "4"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["7", "+", RowBox[List["4", " ", "n"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["4", "+", RowBox[List["2", " ", "n"]]]], "]"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02