Inverse hyperbolic sine: Representations through equivalent functions (formula 01.25.27.0587)

ArcSinh

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving cot^-1

Involving sinh^-1(z)

Involving sinh^-1(z) and cot^-1((-z²)^1/2/(1+z²)^1/2)

http://functions.wolfram.com/01.25.27.0587.01

Input Form

ArcSinh[z] == Sqrt[-z^(-2)] ((Pi z)/2) + (Sqrt[-z^2]/z) ArcCot[Sqrt[-z^2]/Sqrt[1 + z^2]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", RowBox[List["-", "2"]]]]]], FractionBox[RowBox[List["\[Pi]", " ", "z"]], "2"]]], "+", RowBox[List[FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"], RowBox[List["ArcCot", "[", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List["1", "+", 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> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <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> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccot /> <apply> <times /> <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> <power /> <apply> <power /> <apply> <plus /> <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> <apply> <times /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </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["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "z"]], ")"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ArcCot", "[", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], "]"]]]], "z"]]]]]]]

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

2003-08-21