Inverse cosecant: Representations through equivalent functions (formula 01.17.27.1237)

ArcCsc

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving cosh^-1

Involving csc^-1((c z)^1/2)

Involving csc^-1((-z)^1/2) and cosh^-1(1+2/z)

http://functions.wolfram.com/01.17.27.1237.01

Input Form

ArcCsc[Sqrt[-z]] == (-(Sqrt[-z]/2)) Sqrt[1/z] ArcCosh[1 + 2/z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox[RowBox[List["-", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", "z"]]], "2"]]], SqrtBox[FractionBox["1", "z"]], RowBox[List["ArcCosh", "[", RowBox[List["1", "+", FractionBox["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> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 2 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccosh /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <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["ArcCsc", "[", SqrtBox[RowBox[List["-", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", SqrtBox[RowBox[List["-", "z"]]]]], ")"]], " ", SqrtBox[FractionBox["1", "z"]], " ", RowBox[List["ArcCosh", "[", RowBox[List["1", "+", FractionBox["2", "z"]]], "]"]]]]]]]]

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

2003-08-21