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ArcSinh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Representations through equivalent functions > With related functions > Involving sin-1 > Involving sinh-1(a-z/2z1/2) > Involving sinh-1(-z-1/2 z1/2) and sin-1(1/z)





http://functions.wolfram.com/01.25.27.0024.02









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "z"]], "-", "1"]], RowBox[List["2", " ", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], "z"]]]], RowBox[List["2", " ", SqrtBox[FractionBox[RowBox[List["z", "+", "1"]], "z"]]]]], RowBox[List["(", RowBox[List[RowBox[List["ArcSin", "[", FractionBox["1", "z"], "]"]], "+", FractionBox["\[Pi]", "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> <msqrt> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <arcsin /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2001-10-29