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ArcSin




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Representations through equivalent functions > With related functions > Involving sech-1 > Involving sin-1(2 (z2-1)1/2/z2) > Involving sin-1(2 (z2-1)1/2/z2) and sech-1(z)





http://functions.wolfram.com/01.12.27.2271.01









  


  








Input Form





ArcSin[(2 Sqrt[z^2 - 1])/z^2] == (Sqrt[z^2]/z) Pi - 2 ((Sqrt[z^2] Sqrt[z - 1])/(z Sqrt[1 - z])) ArcSech[z] /; Abs[z] > Sqrt[2] && Im[z] != 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]], SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "\[Pi]"]], "-", RowBox[List["2", " ", FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], SqrtBox[RowBox[List["z", "-", "1"]]]]], RowBox[List["z", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", RowBox[List["ArcSech", "[", "z", "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", SqrtBox["2"]]], "\[And]", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[NotEqual]", "0"]]]]]]]]








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> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsech /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <neq /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]], SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", "\[Pi]"]], "z"], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SqrtBox[RowBox[List["z", "-", "1"]]]]], ")"]], " ", RowBox[List["ArcSech", "[", "z", "]"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", SqrtBox["2"]]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[NotEqual]", "0"]]]]]]]]]]


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





2003-08-21