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JacobiSN






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.36.06.0009.01









  


  










Input Form





JacobiSN[z, m] \[Proportional] ((-1)^r/Sqrt[m]) (1/(z - Subscript[z, 0]) + (1/6) (1 + m) (z - Subscript[z, 0]) + (1/360) (7 - 22 m + 7 m^2) (z - Subscript[z, 0])^3 + \[Ellipsis]) /; (z -> Subscript[z, 0]) && Subscript[z, 0] == 2 r EllipticK[m] + (2 s + 1) I EllipticK[1 - m] && Element[r, Integers] && Element[s, Integers]










Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 22 </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> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </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> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <apply> <power /> <apply> <power /> <ci> m </ci> <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 /> <ci> m </ci> <cn type='integer'> 1 </cn> </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'> 7 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 22 </cn> <ci> m </ci> </apply> </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 /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <imaginaryi /> <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





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





2007-05-02