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ArcSin




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.12.27.2018.01









  


  








Input Form





ArcSin[z] == (Sqrt[-z^4]/(2 z^2)) ArcCsch[1/(2 z Sqrt[z^2 - 1])] /; Pi/4 <= Abs[Arg[z]] <= (3 Pi)/4








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "4"]]]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], RowBox[List["ArcCsch", "[", FractionBox["1", RowBox[List["2", " ", "z", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], "]"]]]]]], "/;", RowBox[List[FractionBox["\[Pi]", "4"], "\[LessEqual]", RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox[RowBox[List["3", "\[Pi]"]], "4"]]]]]]]








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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> &#8804; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccsch /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <abs /> <apply> <arg /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "4"]]]], " ", RowBox[List["ArcCsch", "[", FractionBox["1", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], "/;", RowBox[List[FractionBox["\[Pi]", "4"], "\[LessEqual]", RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox[RowBox[List["3", " ", "\[Pi]"]], "4"]]]]]]]]]


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





2003-08-21