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ArcSec




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSec[z] > Representations through equivalent functions > With related functions > Involving sinh-1 > Involving sec-1((2z/(z-(1+z2)1/2))1/2) > Involving sec-1((2z/(z-(1+z2)1/2))1/2) and sinh-1(1/z)





http://functions.wolfram.com/01.18.27.1161.01









  


  








Input Form





ArcSec[Sqrt[(2 z)/(z - Sqrt[1 + z^2])]] == (1/2) I ArcSinh[1/z] /; Pi/2 < Arg[z] < Pi || (Element[(-I) z, Reals] && 0 < (-I) z < 1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSec", "[", SqrtBox[FractionBox[RowBox[List["2", "z"]], RowBox[List["z", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox["1", "z"], "]"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "z", "]"]], "<", "\[Pi]"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "\[Element]", "Reals"]], "\[And]", RowBox[List["0", "<", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsec /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </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> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arg /> <ci> z </ci> </apply> <pi /> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <ci> &#8477; </ci> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSec", "[", SqrtBox[FractionBox[RowBox[List["2", " ", "z_"]], RowBox[List["z_", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox["1", "z"], "]"]]]], "/;", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "z", "]"]], "<", "\[Pi]"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "\[Element]", "Reals"]], "&&", RowBox[List["0", "<", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "<", "1"]]]], ")"]]]]]]]]]]


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





2003-08-21