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ArcCos




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCos[z] > Representations through equivalent functions > With related functions > Involving tanh-1 > Involving cos-1(z) > Involving cos-1(z) and tanh-1(1-2z2/2 z (z2-1)1/2)





http://functions.wolfram.com/01.13.27.1573.01









  


  








Input Form





ArcCos[z] == -(Pi/4) - (I/2) ArcTanh[(1 - 2 z^2)/(2 z Sqrt[z^2 - 1])] /; Element[z, Reals] && z > 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], "-", RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["1", "-", RowBox[List["2", SuperscriptBox["z", "2"]]]]], RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], "]"]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List["z", ">", "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> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> &#8520; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <plus /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctanh /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <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> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> z </ci> <ci> &#8477; </ci> </apply> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCos", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List["z", ">", "1"]]]]]]]]]]


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





2003-08-21