Inverse hyperbolic cosecant: Representations through equivalent functions (formula 01.29.27.1376)

ArcCsch

$Mathematica Notation$

Elementary Functions

ArcCsch[z]

Representations through equivalent functions

With related functions

Involving sinh^-1

Involving csch^-1((-2z)^1/2/(z+a)^1/2)

Involving csch^-1((-2z)^1/2/(z-1)^1/2) and sinh^-1(i/z)

http://functions.wolfram.com/01.29.27.1376.01

Input Form

ArcCsch[Sqrt[-2 z]/Sqrt[z - 1]] == (-(1/2)) ArcSinh[I/z] + (Pi I)/4 /; Inequality[0, Less, Arg[z], LessEqual, Pi]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "2"]], "z"]]], SqrtBox[RowBox[List["z", "-", "1"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["ArcSinh", "[", FractionBox["\[ImaginaryI]", "z"], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]]]]], "/;", RowBox[List["0", "<", 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> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> ⅈ </mi> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mi> π </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccsch /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsinh /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> 0 </cn> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "2"]], " ", "z_"]]], SqrtBox[RowBox[List["z_", "-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["ArcSinh", "[", FractionBox["\[ImaginaryI]", "z"], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]]], "/;", RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]

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

2003-08-21