JacobiNC[u + v, m] == (1 - m JacobiSN[u, m]^2 JacobiSN[v, m]^2)/ (JacobiCN[u, m] JacobiCN[v, m] - JacobiSN[u, m] JacobiDN[u, m] JacobiSN[v, m] JacobiDN[v, m])

Cell[BoxData[RowBox[List[RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List["u", "+", "v"]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["u", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["v", ",", "m"]], "]"]], "2"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["v", ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["v", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["v", ",", "m"]], "]"]]]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> nc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> u </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mrow> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> u </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> u </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> u </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiNC </ci> <apply> <plus /> <ci> u </ci> <ci> v </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> u </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> v </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> u </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> v </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiSN </ci> <ci> u </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> u </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> v </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> v </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List["u_", "+", "v_"]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["u", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["v", ",", "m"]], "]"]], "2"]]]]], RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["v", ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["v", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["v", ",", "m"]], "]"]]]]]]]]]]]

2001-10-29