Inverse hyperbolic cosecant: Representations through equivalent functions (formula 01.29.27.0894)

ArcCsch

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving csc^-1

Involving csch^-1(z)

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

http://functions.wolfram.com/01.29.27.0894.01

Input Form

ArcCsch[z] == (-(z/Sqrt[-z^2])) ArcCsc[Sqrt[-z^2]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["ArcCsc", "[", SqrtBox[RowBox[List["-", SuperscriptBox["z", "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> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsch /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arccsc /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["z", " ", RowBox[List["ArcCsc", "[", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "]"]]]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]]]]]]

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

2003-08-21