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JacobiSN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSN[z,m] > Identities > Functional identities





http://functions.wolfram.com/09.36.17.0002.01









  


  








Input Form





JacobiSN[z, m] == ((Sqrt[\[Mu]] + 1) JacobiSN[z/(Sqrt[\[Mu]] + 1), \[Mu]])/ (Sqrt[\[Mu]] JacobiSN[z/(Sqrt[\[Mu]] + 1), \[Mu]]^2 + 1) /; \[Mu] == (1 - Sqrt[1 - m])^4/m^2 && 0 < m < 1








Standard Form





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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> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> &#956; </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mrow> <msqrt> <mi> &#956; </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#10072; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <msqrt> <mi> &#956; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mrow> <msqrt> <mi> &#956; </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#10072; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#956; </mi> <mo> &#10869; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> m </mi> <mo> &lt; </mo> <mn> 1 </mn> </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> <power /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> JacobiSN </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#956; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> &#956; </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Mu]"], "+", "1"]], ")"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[FractionBox["z", RowBox[List[SqrtBox["\[Mu]"], "+", "1"]]], ",", "\[Mu]"]], "]"]]]], RowBox[List[RowBox[List[SqrtBox["\[Mu]"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox["z", RowBox[List[SqrtBox["\[Mu]"], "+", "1"]]], ",", "\[Mu]"]], "]"]], "2"]]], "+", "1"]]], "/;", RowBox[List[RowBox[List["\[Mu]", "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]], "4"], SuperscriptBox["m", "2"]]]], "&&", RowBox[List["0", "<", "m", "<", "1"]]]]]]]]]]


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





2001-10-29