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JacobiSN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSN[z,m] > Representations through equivalent functions > With related functions > Involving Weierstrass functions





http://functions.wolfram.com/09.36.27.0024.01









  


  










Input Form





JacobiSN[z, m] == (1/Sqrt[m]) (WeierstrassZeta[z + Subscript[\[Omega], 1] + Subscript[\[Omega], 3], {Subscript[g, 2], Subscript[g, 3]}] - WeierstrassZeta[z + Subscript[\[Omega], 3], {Subscript[g, 2], Subscript[g, 3]}] - WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}]) /; {Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == {2 EllipticK[m], -2 EllipticK[m] - I EllipticK[1 - m]} && {Subscript[g, 2], Subscript[g, 3]} == WeierstrassInvariants[ {Subscript[\[Omega], 1], Subscript[\[Omega], 3]}]










Standard Form





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MathML Form







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</mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> WeierstrassZeta </ci> <apply> <plus /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> WeierstrassZeta </ci> <apply> <ci> Subscript </ci> <ci> &#969; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List[RowBox[List["z", "+", SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "-", RowBox[List["WeierstrassZeta", "[", RowBox[List[RowBox[List["z", "+", SubscriptBox["\[Omega]", "3"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "-", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], SqrtBox["m"]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], ",", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]], "}"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]]]]]]]]]]










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





2001-10-29





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