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 cos-1 > Involving tan-1(2 (z2-1)1/2/-2+z2) > Involving tan-1(2 (z2-1)1/2/-2+z2) and cos-1(1/z)





http://functions.wolfram.com/01.14.27.0615.01









  


  










Input Form





ArcTan[(2 Sqrt[z^2 - 1])/(-2 + z^2)] == (Pi/(2 Sqrt[-1 + z^2])) (2 Sqrt[1 - 1/z^2] z + (-2 + z^2) Sqrt[z^4/(-1 + z^2)] Sqrt[(-1 + z^2)/z^4] Sqrt[(-1 + z^2)/(-2 + z^2)^2] - Sqrt[1 - 1/z^2] z (Sqrt[I/z] Sqrt[(-I) z] - Sqrt[-(I/z)] Sqrt[I z] + Sqrt[1/z^2] z - Sqrt[(-1 + z)/z] Sqrt[z/(-1 + z)] + Sqrt[1 + 1/z] Sqrt[z/(1 + z)])) - ((2 z)/Sqrt[-1 + z^2]) Sqrt[1 - 1/z^2] ArcCos[1/z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]], RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["z", "2"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox["z", "4"], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["z", "2"]]], ")"]], "2"]]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], "-", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]]]], "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z"]], "-", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]], " ", SqrtBox[FractionBox["z", RowBox[List["1", "+", "z"]]]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["2", "z"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], RowBox[List["ArcCos", "[", FractionBox["1", "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> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctan /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <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 /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <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='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccos /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2003-08-21