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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Series representations > Generalized power series > Expansions at z==-1 > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.30.06.0093.01









  


  








Input Form





ArcSech[z]^2 == Subscript[F, Infinity][z] /; Subscript[F, n][z] == -(Pi - 2 Sqrt[-z - 1] Sum[(Pochhammer[1/2, k]/k!) Hypergeometric2F1[1/2, 3/2 + k, 3/2, -1] (z + 1)^k, {k, 0, n}])^2 == -(ArcSec[z] + ((2 Pochhammer[1/2, n + 1])/(n + 1)!) (z + 1)^(n + 1) Sqrt[-z - 1] HypergeometricPFQ[{{5/2 + n}, {1/2}, {1, 3/2 + n}}, {{}, {3/2}, {2 + n, 5/2 + n}}, -1, z + 1])^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["ArcSech", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "z"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], ")"]], "2"]]], "\[Equal]", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcSec", "[", "z", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "z"]], "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "2"], "+", "n"]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "1"]], ",", RowBox[List["z", "+", "1"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], ")"]], "2"]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02