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Tanh




Mathematica Notation

Traditional Notation









Elementary Functions > Tanh[z] > Transformations > Transformations and argument simplifications > Argument involving inverse trigonometric and hyperbolic functions > Involving tan-1





http://functions.wolfram.com/01.21.16.0072.01









  


  








Input Form





Tanh[a ArcTan[x, y]] == -1 + 2/(1 + ((x + I y)/Sqrt[x^2 + y^2])^(2 I a))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Tanh", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["2", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 2 </mn> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <tanh /> <apply> <times /> <ci> a </ci> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Tanh", "[", RowBox[List["a_", " ", RowBox[List["ArcTan", "[", RowBox[List["x_", ",", "y_"]], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["2", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]]]]]]]]]]


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





2007-05-02