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
 











variants of this functions
ArcTan




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[z] > Representations through equivalent functions > With related functions > Involving tanh-1 > Involving tan-1(1-z/1+z) > Involving tan-1(1-z/1+z) and tanh-1(i z)





http://functions.wolfram.com/01.14.27.1964.01









  


  








Input Form





ArcTan[(1 - z)/(1 + z)] == I ArcTanh[I z] - (Pi/4) (1 - 2 Sqrt[1 - I z] Sqrt[1/(1 - I z)] - (-1 + Sqrt[z^2]/z) (1 + ((1 - z^2)/z) Sqrt[z^2/(-1 + z^2)^2])) /; Abs[z] != 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "-", RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "z"], " ", SqrtBox[FractionBox[SuperscriptBox["z", "2"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "2"]]]]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[NotEqual]", "1"]]]]]]








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> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </msqrt> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arctan /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <arctanh /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <neq /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["1", "-", "z_"]], RowBox[List["1", "+", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox["z", "2"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], "2"]]]]], "z"]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[NotEqual]", "1"]]]]]]]]


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





2003-08-21