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ArcSinh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/01.25.06.0026.01









  


  










Input Form





ArcSinh[z] == -((Pi I)/2) + (1/(1 - I Subscript[z, 0]))^ ((1/2) Floor[Arg[I (Subscript[z, 0] - z)]/(2 Pi)]) (1 - I Subscript[z, 0])^((1/2) Floor[Arg[I (Subscript[z, 0] - z)]/(2 Pi)]) (2 Pi I^Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)] Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)] Floor[(Pi + Arg[1 + I Subscript[z, 0]])/(2 Pi)] + (1/2) (1/(1 + I Subscript[z, 0]))^ ((1/2) Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)]) (1 + I Subscript[z, 0])^((1/2) Floor[Arg[I (z - Subscript[z, 0])]/ (2 Pi)]) (Pi I + Sqrt[Pi] Sum[((2^k Subscript[z, 0]^(1 - k))/k!) HypergeometricPFQRegularized[{1/2, 1/2, 1}, {1 - k/2, (3 - k)/2}, -Subscript[z, 0]^2] (z - Subscript[z, 0])^k, {k, 0, Infinity}]))










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "k"], " ", SubsuperscriptBox["zz", "0", RowBox[List["1", "-", "k"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["k", "2"]]], ",", FractionBox[RowBox[List["3", "-", "k"]], "2"]]], "}"]], ",", RowBox[List["-", SubsuperscriptBox["zz", "0", "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]]]], ")"]]]]]]]]]]










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





2007-05-02





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