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

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

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2001-10-29