Inverse cosecant: Representations through equivalent functions (formula 01.17.27.1052)

ArcCsc

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving sinh^-1

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

Involving csc^-1(1/z^1/2) and sinh^-1((-z)^1/2)

http://functions.wolfram.com/01.17.27.1052.01

Input Form

ArcCsc[1/Sqrt[z]] == (Sqrt[z]/Sqrt[-z]) ArcSinh[Sqrt[-z]]

Standard Form

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

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

2003-08-21