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ArcSech






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Representations through more general functions > Through hypergeometric functions of two variables





http://functions.wolfram.com/01.30.26.0009.01









  


  










Input Form





ArcSech[z] == I (2 UnitStep[Im[1/z]] - 1) (Pi - Sqrt[2] Sqrt[-z - 1] HypergeometricPFQ[{{3/2}, {1/2}, {1/2}}, {{}, {3/2}, {3/2}}, -1, z + 1]) /; !IntervalMemberQ[Interval[{0, 1}], z]










Standard Form





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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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> &#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 0 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sech </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#8520; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> &#952; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> Im </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> z </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SqrtBox </ci> <ms> 2 </ms> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> 1 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> 1 </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> </list> </apply> </list> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> </list> </apply> </apply> </list> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> z </ms> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> &#8713; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> , </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", FractionBox["1", "z"], "]"]], "]"]]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "z"]], "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]]]], "}"]], ",", RowBox[List["-", "1"]], ",", RowBox[List["z", "+", "1"]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]










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





2001-10-29





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