Inverse cosecant: Representations through equivalent functions (formula 01.17.27.1292)

ArcCsc

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving cosh^-1

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

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

http://functions.wolfram.com/01.17.27.1292.01

Input Form

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

Standard Form

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

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsc", "[", SqrtBox[FractionBox["2", RowBox[List["1", "+", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["z", "+", "1"]]], " ", SqrtBox[FractionBox["1", RowBox[List["z", "+", "1"]]]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["ArcCosh", "[", "z", "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]]], ")"]]]]]]]]

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

2003-08-21