Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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
 











JacobiNC




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiNC[z,m] > Series representations > Generalized power series > Expansions at z==(2 r+1) K(m)+2 i s K(1-m)





http://functions.wolfram.com/09.31.06.0008.01









  


  








Input Form





JacobiNC[z, m] \[Proportional] ((-1)^(r + s - 1)/Sqrt[1 - m]) (1/(z - Subscript[z, 0]) + (1/6) (1 - 2 m) (z - Subscript[z, 0]) + (1/360) (7 + 8 m - 8 m^2) (z - Subscript[z, 0])^3 + \[Ellipsis]) /; (z -> Subscript[z, 0]) && Subscript[z, 0] == (2 r + 1) EllipticK[m] + 2 s I EllipticK[1 - m] && Element[r, Integers] && Element[s, Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "s", "-", "1"]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " "]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["z", "-", SubscriptBox["z", "0"]]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "360"], " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["8", " ", "m"]], "-", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "r"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["2", "s", " ", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["s", "\[Element]", "Integers"]]]]]]]]








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> <mi> nc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> <mtext> </mtext> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 360 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#63449; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> s </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> r </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> JacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> r </ci> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 360 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -8 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> m </ci> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> s </ci> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> r </ci> <integers /> </apply> <apply> <in /> <ci> s </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "s", "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["z", "-", SubscriptBox["zz", "0"]]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "360"], " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["8", " ", "m"]], "-", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "3"]]], "+", "\[Ellipsis]"]], ")"]]]], SqrtBox[RowBox[List["1", "-", "m"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["2", " ", "s", " ", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["s", "\[Element]", "Integers"]]]]]]]]]]


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





2007-05-02