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JacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations





http://functions.wolfram.com/09.29.16.0013.01









  


  










Input Form





JacobiDN[(1 + Sqrt[m]) z, (4 Sqrt[m])/(1 + Sqrt[m])^2] == (1 - Sqrt[m] JacobiSN[z, m]^2)/(1 + Sqrt[m] JacobiSN[z, m]^2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["m"]]], ")"]], " ", "z"]], ",", FractionBox[RowBox[List["4", SqrtBox["m"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["m"]]], ")"]], "2"]]]], "]"]], "\[Equal]", FractionBox[RowBox[List["1", "-", RowBox[List[SqrtBox["m"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]], RowBox[List["1", "+", RowBox[List[SqrtBox["m"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </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> <ci> JacobiDN </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["m_"]]], ")"]], " ", "z_"]], ",", FractionBox[RowBox[List["4", " ", SqrtBox["m_"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["m_"]]], ")"]], "2"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["1", "-", RowBox[List[SqrtBox["m"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]], RowBox[List["1", "+", RowBox[List[SqrtBox["m"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]]]]]]]










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





2001-10-29