Inverse hyperbolic cosecant: Representations through equivalent functions (formula 01.29.27.1298)

ArcCsch

$Mathematica Notation$

Elementary Functions

ArcCsch[z]

Representations through equivalent functions

With related functions

Involving sinh^-1

Involving csch^-1(a (b z^c)^m)

Involving csch^-1(a (b z^c)^m) and sinh^-1(1/a b^-m z^{-m c})

http://functions.wolfram.com/01.29.27.1298.01

Input Form

ArcCsch[a (b z^c)^m] == ((b z^c)^m/(b^m z^(m c))) ArcSinh[1/a/(b^m z^(m c))] /; Element[2 m, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"], " "]], RowBox[List[SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]], RowBox[List["ArcSinh", "[", RowBox[List[FractionBox["1", "a"], " ", SuperscriptBox["b", RowBox[List["-", "m"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "m"]], " ", "c"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["2", " ", "m"]], "\[Element]", "Integers"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mtext> </mtext> </mrow> <mrow> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> a </mi> </mfrac> <mo> ⁢ </mo> <msup> <mi> b </mi> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccsch /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsinh /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", RowBox[List["a_", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b_", " ", SuperscriptBox["z_", "c_"]]], ")"]], "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"], " ", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[SuperscriptBox["b", RowBox[List["-", "m"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "m"]], " ", "c"]]]]], "a"], "]"]]]], RowBox[List[SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]], "/;", RowBox[List[RowBox[List["2", " ", "m"]], "\[Element]", "Integers"]]]]]]]]

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

2003-08-21