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ArcCsch




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsch[z] > Representations through equivalent functions > With related functions > Involving sec-1 > Involving csch-1((z2)1/2) > Involving csch-1((z2)1/2) and sec-1(i z)





http://functions.wolfram.com/01.29.27.1110.01









  


  








Input Form





ArcCsch[Sqrt[z^2]] == I ArcSec[I z] - (Pi I)/2 /; Inequality[Pi/2, Less, Arg[z], LessEqual, Pi] || Inequality[-Pi, Less, Arg[z], LessEqual, -(Pi/2)]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]]]], "/;", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "\[Or]", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]]]]]]








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> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </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> &#8804; </mo> <mi> &#960; </mi> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccsch /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <arcsec /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <ci> Inequality </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <pi /> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", SqrtBox[SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "/;", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "||", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]]]]]]]]


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





2003-08-21