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ArcSin






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Series representations > Generalized power series > Expansions at z==infinity > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.12.06.0065.01









  


  










Input Form





ArcSin[z]^2 \[Proportional] (-(1/4)) Log[-z^2]^2 + Log[-z^2] (-Log[2] + 1/(4 z^2) + 3/(32 z^4) + \[Ellipsis]) + (-Log[2]^2 + Log[2]/(2 z^2) + (3 Log[2] - 1)/(16 z^4) + \[Ellipsis]) /; (Abs[z] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSin", "[", "z", "]"]], "2"], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], "2"]]], "+", " ", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox["1", RowBox[List["4", SuperscriptBox["z", "2"]]]], "+", FractionBox["3", RowBox[List["32", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"]]], "+", FractionBox[RowBox[List["Log", "[", "2", "]"]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["3", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", "1"]], RowBox[List["16", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]










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> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8733; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <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> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <arcsin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <cn type='integer'> 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> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcSin", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox["3", RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"]]], "+", FractionBox[RowBox[List["Log", "[", "2", "]"]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["3", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", "1"]], RowBox[List["16", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02





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