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ArcSin




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.12.27.0202.01









  


  








Input Form





ArcSin[Sqrt[z^2 - 1]/z] == (Pi/2) (Sqrt[z^2]/z - 1) + (Sqrt[z - 1]/Sqrt[1 - z]) ArcCosh[1/z] /; Re[z] != 0 && Abs[Arg[z]] < Pi








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "z"], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["(", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox[SqrtBox[RowBox[List["z", "-", "1"]]], SqrtBox[RowBox[List["1", "-", "z"]]]], " ", RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[NotEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "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> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <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> &lt; </mo> <mtext> </mtext> <mi> &#960; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <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 /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </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> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <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> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccosh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <neq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <ci> z </ci> </apply> </apply> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], "z_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", "1"]], ")"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " ", RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[NotEqual]", "0"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]]


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





2001-10-29