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ArcSec




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSec[z] > Series representations > Generalized power series > Expansions at z==infinity > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.18.06.0067.01









  


  








Input Form





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


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





2007-05-02