Inverse hyperbolic cosecant: Representations through equivalent functions (formula 01.29.27.2190)

ArcCsch

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving sech^-1

Involving csch^-1((c z²)^1/2)

Involving csch^-1((z₂)^1/2) and sech^-1(i z)

http://functions.wolfram.com/01.29.27.2190.01

Input Form

ArcCsch[Sqrt[z^2]] == Sqrt[-1 + I z] Sqrt[1/(-1 + I z)] ArcSech[I z] + (Pi I Sqrt[z^2])/(2 z)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], RowBox[List["ArcSech", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SqrtBox[SuperscriptBox["z", "2"]]]], RowBox[List["2", " ", "z"]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> ⩵ </mo> <mrow> <mrow> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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 /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsech /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <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["ArcCsch", "[", SqrtBox[SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", RowBox[List["ArcSech", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SqrtBox[SuperscriptBox["z", "2"]]]], RowBox[List["2", " ", "z"]]]]]]]]]

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

2003-08-21