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ArcCos






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCos[z] > Series representations > Generalized power series > Expansions at z==1 > For the function itself





http://functions.wolfram.com/01.13.06.0044.01









  


  










Input Form





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










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





2007-05-02





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