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ArcTanh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.27.06.0023.01









  


  










Input Form





ArcTanh[z] == ArcTanh[Subscript[z, 0]] + (1/2) (Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] (Log[1 + Subscript[z, 0]] + Log[1/(1 + Subscript[z, 0])]) - Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] (Log[1 - Subscript[z, 0]] + Log[1/(1 - Subscript[z, 0])]) + Sum[(((-1)^(k - 1)/(1 + Subscript[z, 0])^k + (1 - Subscript[z, 0])^(-k))/ k) (z - Subscript[z, 0])^k, {k, 1, Infinity}])










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcTanh", "[", SubscriptBox["zz", "0"], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]], "+", RowBox[List["Log", "[", FractionBox["1", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], "]"]], "+", RowBox[List["Log", "[", FractionBox["1", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], "]"]]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["-", "k"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["-", "k"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], "k"]]]]], ")"]]]]]]]]]]










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





2007-05-02