Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArcCot






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCot[z] > Continued fraction representations





http://functions.wolfram.com/01.16.10.0001.01









  


  










Input Form





ArcCot[z] == 1/ (z (1 + 1/(z^2 (3 + 4/(z^2 (5 + 9/(z^2 (7 + 16/(z^2 (9 + 25/(z^2 (11 + 36/(z^2 (13 + \[Ellipsis])))))))))))))) /; !IntervalMemberQ[Interval[{-1, 1}], I z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCot", "[", "z", "]"]], "\[Equal]", " ", FractionBox[SuperscriptBox["z", RowBox[List["-", "1"]]], RowBox[List["1", "+", FractionBox[SuperscriptBox["z", RowBox[List["-", "2"]]], RowBox[List["3", "+", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["5", "+", FractionBox[RowBox[List["9", " ", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["7", "+", FractionBox[RowBox[List["16", " ", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["9", "+", FractionBox[RowBox[List["25", " ", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["11", "+", FractionBox[RowBox[List["36", " ", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["13", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]]], "/;", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mtext> </mtext> <mfrac> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 25 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 36 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 13 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mtext> &#8203; </mtext> <mi> z </mi> </mrow> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mtext> </mtext> <mfrac> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 25 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 36 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 13 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mtext> &#8203; </mtext> <mi> z </mi> </mrow> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCot", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["1", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["3", "+", FractionBox["4", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["5", "+", FractionBox["9", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["7", "+", FractionBox["16", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["9", "+", FractionBox["25", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["11", "+", FractionBox["36", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["13", "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]]










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





2001-10-29