Inverse hyperbolic cosecant: Series representations (formula 01.29.06.0033)

ArcCsch

$Mathematica Notation$

Elementary Functions

ArcCsch[z]

Series representations

Generalized power series

Expansions on branch cuts

For the function itself

In the upper half-plane

http://functions.wolfram.com/01.29.06.0033.01

Input Form

ArcCsch[z] == (Pi I)/2 - 2 Pi I^Floor[Arg[(-I) (z - x)]/(2 Pi)] Floor[Arg[(-I) (z - x)]/(2 Pi)] - Exp[Pi I Floor[Arg[(-I) (z - x)]/(2 Pi)]] ((Pi I)/2 + Sum[(-1)^(k - 1) x^(-k - 1) HypergeometricPFQ[ {1/2, (1 + k)/2, 1 + k/2}, {1, 3/2}, -(1/x^2)] (z - x)^k, {k, 0, Infinity}]) /; Element[I x, Reals] && -1 < I x < 0

Standard Form

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MathML Form

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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], "-", RowBox[List["2", " ", "\[Pi]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", "k"]], "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["1", "+", FractionBox["k", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["\[ImaginaryI]", " ", "x"]], "<", "0"]]]]]]]]]]

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

2007-05-02