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ArcCosh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCosh[z] > Representations through equivalent functions > With related functions > Involving tan-1 > Involving cosh-1(1+z/1-z) > Involving cosh-1(1+z/1-z) and tan-1((-1/z)1/2)





http://functions.wolfram.com/01.26.27.0539.01









  


  








Input Form





ArcCosh[(1 + z)/(1 - z)] == (-Pi) I + 2 I ArcTan[Sqrt[-(1/z)]] /; Im[z] < 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", FractionBox[RowBox[List["1", "+", "z"]], RowBox[List["1", "-", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]"]], "+", RowBox[List["2", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "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> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccosh /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <power /> <apply> <times /> <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> </apply> </apply> </apply> <apply> <lt /> <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["ArcCosh", "[", FractionBox[RowBox[List["1", "+", "z_"]], RowBox[List["1", "-", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "0"]]]]]]]]


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





2003-08-21