Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











EllipticK






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticK[z] > Specific values > Singular values





http://functions.wolfram.com/08.02.03.0020.01









  


  










Input Form





EllipticK[1 - z^2]/EllipticK[z^2] == Sqrt[11] /; z == (1/2) Sqrt[(32 3^(2/3) + 3 (9 + 7 Sqrt[33])^(1/3) - 4 3^(1/3) (9 + 7 Sqrt[33])^(2/3))/(6 (9 + 7 Sqrt[33])^(1/3) + Sqrt[3 (-32 3^(2/3) (9 + 7 Sqrt[33])^(1/3) + 9 (9 + 7 Sqrt[33])^(2/3) + 4 3^(1/3) (9 + 7 Sqrt[33]))])]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticK", "[", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["EllipticK", "[", SuperscriptBox["z", "2"], "]"]]], "\[Equal]", SqrtBox["11"]]], "/;", RowBox[List["z", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["32", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["4", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", SqrtBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "32"]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List["9", " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List["4", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]]]]]], ")"]]]]]]], ")"]]]], ")"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mfrac> <mo> &#10869; </mo> <msqrt> <mn> 11 </mn> </msqrt> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8730; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 32 </mn> </mrow> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 11 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <eq /> <ci> z </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <root /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -32 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["EllipticK", "[", RowBox[List["1", "-", SuperscriptBox["z_", "2"]]], "]"]], RowBox[List["EllipticK", "[", SuperscriptBox["z_", "2"], "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox["11"], "/;", RowBox[List["z", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["32", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["4", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], RowBox[List[RowBox[List["6", " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", SqrtBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "32"]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List["9", " ", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List["4", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["7", " ", SqrtBox["33"]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.