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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)1/2) > Involving tan-1(1/(-z)1/2) and tanh-1(1/z1/2)





http://functions.wolfram.com/01.14.27.1947.01









  


  








Input Form





ArcTan[1/Sqrt[-z]] == (-I) ArcTanh[Sqrt[1/z]] /; Im[z] <= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", FractionBox[StyleBox["1", "Program"], SqrtBox[RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcTanh", "[", SqrtBox[FractionBox[StyleBox["1", "Program"], "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]








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> <mn> 1 </mn> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arctan /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <arctanh /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcTanh", "[", SqrtBox[FractionBox["1", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]]]


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





2003-08-21