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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Representations through equivalent functions > With related functions > Involving tanh-1 > Involving sech-1(z2+1/2z) > Involving sech-1(z2+1/2z) and tanh-1(i/z)





http://functions.wolfram.com/01.30.27.2042.01









  


  








Input Form





ArcSech[(z^2 + 1)/(2 z)] == (3 Pi I)/2 + 2 ArcTanh[I/z] /; Abs[z] <= 1 && Pi/2 <= Arg[z] <= Pi








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", FractionBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]], RowBox[List["2", "z"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]", " ", "\[ImaginaryI]"]], "2"], "+", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", FractionBox["\[ImaginaryI]", "z"], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[LessEqual]", "1"]], "\[And]", RowBox[List[FractionBox["\[Pi]", "2"], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]








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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8804; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsech /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <leq /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <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> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", FractionBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]], RowBox[List["2", " ", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "2"], "+", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", FractionBox["\[ImaginaryI]", "z"], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[LessEqual]", "1"]], "&&", RowBox[List[FractionBox["\[Pi]", "2"], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]]]


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





2003-08-21