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ArcCoth






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.28.06.0022.01









  


  










Input Form





ArcCoth[z] == ArcCoth[Subscript[z, 0]] + (1/2) (Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] (Log[(1 + Subscript[z, 0])/Subscript[z, 0]] + Log[Subscript[z, 0]/(1 + Subscript[z, 0])]) - Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] (Log[(-1 + Subscript[z, 0])/Subscript[z, 0]] + Log[Subscript[z, 0]/(-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







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</mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccoth /> <ci> z </ci> </apply> <apply> <plus /> <apply> <arccoth /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcCoth", "[", SubscriptBox["zz", "0"], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[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", "[", FractionBox[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], SubscriptBox["zz", "0"]], "]"]], "+", RowBox[List["Log", "[", FractionBox[SubscriptBox["zz", "0"], RowBox[List["1", "+", SubscriptBox["zz", "0"]]]], "]"]]]], ")"]]]], "-", 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", "[", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["zz", "0"]]], SubscriptBox["zz", "0"]], "]"]], "+", RowBox[List["Log", "[", FractionBox[SubscriptBox["zz", "0"], RowBox[List[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





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