Inverse hyperbolic sine: Representations through equivalent functions (formula 01.25.27.0239)

ArcSinh

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving cos^-1

Involving sinh^-1((z-a)^1/2/(-2z)^1/2)

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

http://functions.wolfram.com/01.25.27.0239.01

Input Form

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

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List["z", "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["-", "2"]], " ", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["ArcCos", "[", FractionBox["1", "z"], "]"]], "-", "\[Pi]"]], ")"]]]]]]]]

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> <mfrac> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <arccos /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

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

2003-08-21