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JacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/09.29.06.0001.02









  


  










Input Form





JacobiDN[z, m] \[Proportional] SeriesData[$CellContext`z, 0, {1, 0, Rational[1, 6] $CellContext`m, 0, Rational[1, 120] (-4 $CellContext`m + 9 $CellContext`m^2), 0, Rational[1, 5040] (16 $CellContext`m - 180 $CellContext`m^2 + 225 $CellContext`m^3), 0, Rational[1, 362880] (-64 $CellContext`m + 3024 $CellContext`m^2 - 12600 $CellContext`m^3 + 11025 $CellContext`m^4)}, 1, 11, 1]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", InterpretationBox[RowBox[List["1", "-", FractionBox[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], "2"], "+", RowBox[List[FractionBox[RowBox[List["m", RowBox[List["(", RowBox[List["m", "+", "4"]], ")"]]]], "24"], " ", SuperscriptBox["z", "4"]]], "+", RowBox[List[FractionBox["1", "720"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "-", RowBox[List["44", " ", SuperscriptBox["m", "2"]]], "-", SuperscriptBox["m", "3"]]], ")"]], " ", SuperscriptBox["z", "6"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["64", " ", "m"]], "+", RowBox[List["912", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["408", " ", SuperscriptBox["m", "3"]]], "+", SuperscriptBox["m", "4"]]], ")"]], " ", SuperscriptBox["z", "8"]]], "40320"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "256"]], " ", "m"]], "-", RowBox[List["15808", " ", SuperscriptBox["m", "2"]]], "-", RowBox[List["30768", " ", SuperscriptBox["m", "3"]]], "-", RowBox[List["3688", " ", SuperscriptBox["m", "4"]]], "-", SuperscriptBox["m", "5"]]], ")"]], " ", SuperscriptBox["z", "10"]]], "3628800"], "+", RowBox[List["O", "[", SuperscriptBox["z", "12"], "]"]]]], SeriesData[$CellContext`z, 0, List[1, 0, Times[Rational[1, 6], $CellContext`m], 0, Times[Rational[1, 120], Plus[Times[-4, $CellContext`m], Times[9, Power[$CellContext`m, 2]]]], 0, Times[Rational[1, 5040], Plus[Times[16, $CellContext`m], Times[-1, 180, Power[$CellContext`m, 2]], Times[225, Power[$CellContext`m, 3]]]], 0, Times[Rational[1, 362880], Plus[Times[-64, $CellContext`m], Times[3024, Power[$CellContext`m, 2]], Times[-1, 12600, Power[$CellContext`m, 3]], Times[11025, Power[$CellContext`m, 4]]]]], 1, 11, 1]]]]]]










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> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 24 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 720 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 16 </mn> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 44 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mn> 912 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 408 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mn> 40320 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 256 </mn> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 15808 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 30768 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3688 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mn> 3628800 </mn> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 12 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 24 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 720 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -16 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 44 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 64 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 912 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 408 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <cn type='integer'> 40320 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -256 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15808 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30768 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3688 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> <apply> <power /> <cn type='integer'> 3628800 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 12 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", InterpretationBox[RowBox[List["$CellContext`z", "+", RowBox[List[FractionBox["1", "6"], " ", "$CellContext`m", " ", SuperscriptBox["$CellContext`z", "3"]]], "+", RowBox[List[FractionBox["1", "120"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "$CellContext`m"]], "+", RowBox[List["9", " ", SuperscriptBox["$CellContext`m", "2"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "5"]]], "+", RowBox[List[FractionBox["1", "5040"], " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", "$CellContext`m"]], "-", RowBox[List["180", " ", SuperscriptBox["$CellContext`m", "2"]]], "+", RowBox[List["225", " ", SuperscriptBox["$CellContext`m", "3"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "7"]]], "+", RowBox[List[FractionBox["1", "362880"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "64"]], " ", "$CellContext`m"]], "+", RowBox[List["3024", " ", SuperscriptBox["$CellContext`m", "2"]]], "-", RowBox[List["12600", " ", SuperscriptBox["$CellContext`m", "3"]]], "+", RowBox[List["11025", " ", SuperscriptBox["$CellContext`m", "4"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "9"]]], "+", InterpretationBox[SuperscriptBox[RowBox[List["O", "[", "$CellContext`z", "]"]], "11"], SeriesData[$CellContext`z, 0, List[], 1, 11, 1], Rule[Editable, False]]]], SeriesData[$CellContext`z, 0, List[1, 0, Times[Rational[1, 6], $CellContext`m], 0, Times[Rational[1, 120], Plus[Times[-4, $CellContext`m], Times[9, Power[$CellContext`m, 2]]]], 0, Times[Rational[1, 5040], Plus[Times[16, $CellContext`m], Times[-1, 180, Power[$CellContext`m, 2]], Times[225, Power[$CellContext`m, 3]]]], 0, Times[Rational[1, 362880], Plus[Times[-64, $CellContext`m], Times[3024, Power[$CellContext`m, 2]], Times[-1, 12600, Power[$CellContext`m, 3]], Times[11025, Power[$CellContext`m, 4]]]]], 1, 11, 1], Rule[Editable, False]]]]]]










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





2001-10-29





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