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Sqrt




Mathematica Notation

Traditional Notation









Elementary Functions > Sqrt[z] > Continued fraction representations





http://functions.wolfram.com/01.01.10.0005.01









  


  








Input Form





Sqrt[1 + z] == 1 + z/ (4 (1/2 + z/(16 (1/2 + z/(16 (1/2 + z/(16 (1/2 + z/(16 \[Ellipsis]))))))))\ )








Standard Form





Cell[BoxData[RowBox[List[SqrtBox[RowBox[List["1", "+", "z"]]], "\[Equal]", RowBox[List["1", "+", FractionBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", "\[Ellipsis]"]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> z </mi> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> z </mi> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> z </mi> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> z </mi> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mo> &#8230; </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SqrtBox[RowBox[List["1", "+", "z_"]]], "]"]], "\[RuleDelayed]", RowBox[List["1", "+", FractionBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["z", RowBox[List["16", " ", "\[Ellipsis]"]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]]]]]]










Contributed by





Andreas Lauschke (2006)


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





2007-05-02