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http://functions.wolfram.com/11.04.06.0007.01
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MathieuSPrime[MathieuCharacteristicB[4, q], q, z] \[Proportional]
4 Cos[4 z] + ((1/6) Cos[2 z] - (3/10) Cos[6 z]) q +
((-(17/900)) Cos[4 z] + (1/120) Cos[8 z]) q^2 +
((-(1/800)) Cos[2 z] + (29 Cos[6 z])/48000 - Cos[10 z]/8064) q^3 +
((18839 Cos[4 z])/103680000 - Cos[8 z]/112000 + Cos[12 z]/860160) q^4 +
((40159 Cos[2 z])/2488320000 - (65633 Cos[6 z])/9676800000 +
(67 Cos[10 z])/928972800 - Cos[14 z]/132710400) q^5 +
(-((1933949 Cos[4 z])/677376000000) + (23351 Cos[8 z])/199065600000 -
Cos[12 z]/3096576000 + Cos[16 z]/27869184000) q^6 +
SeriesData[$CellContext`q, 0, {}, 0, 11, 1]
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Cell[BoxData[RowBox[List[RowBox[List["MathieuSPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List["4", ",", "q"]], "]"]], ",", "q", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["4", " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List[FractionBox["3", "10"], " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]]]], ")"]], " ", "q"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["17", "900"]]], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "120"], " ", RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["q", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "800"]]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["29", " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]], "48000"], "-", FractionBox[RowBox[List["Cos", "[", RowBox[List["10", " ", "z"]], "]"]], "8064"]]], ")"]], " ", SuperscriptBox["q", "3"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["18839", " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "103680000"], "-", FractionBox[RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]], "112000"], "+", FractionBox[RowBox[List["Cos", "[", RowBox[List["12", " ", "z"]], "]"]], "860160"]]], ")"]], " ", SuperscriptBox["q", "4"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["40159", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "2488320000"], "-", FractionBox[RowBox[List["65633", " ", RowBox[List["Cos", "[", RowBox[List["6", " ", "z"]], "]"]]]], "9676800000"], "+", FractionBox[RowBox[List["67", " ", RowBox[List["Cos", "[", RowBox[List["10", " ", "z"]], "]"]]]], "928972800"], "-", FractionBox[RowBox[List["Cos", "[", RowBox[List["14", " ", "z"]], "]"]], "132710400"]]], ")"]], " ", SuperscriptBox["q", "5"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["1933949", " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "z"]], "]"]]]], "677376000000"]]], "+", FractionBox[RowBox[List["23351", " ", RowBox[List["Cos", "[", RowBox[List["8", " ", "z"]], "]"]]]], "199065600000"], "-", FractionBox[RowBox[List["Cos", "[", RowBox[List["12", " ", "z"]], "]"]], "3096576000"], "+", FractionBox[RowBox[List["Cos", "[", RowBox[List["16", " ", "z"]], "]"]], "27869184000"]]], ")"]], " ", SuperscriptBox["q", "6"]]], "+", InterpretationBox[SuperscriptBox[RowBox[List["O", "[", "q", "]"]], "11"], SeriesData[$CellContext`q, 0, List[], 0, 11, 1]]]]]]]]
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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> <msup> <mi> Se </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> b </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicB </ci> </annotation-xml> </semantics> <mn> 4 </mn> </msub> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mi> q </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <semantics> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 10 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 120 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 17 </mn> <mn> 900 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 800 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 29 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 48000 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 8064 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 18839 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 103680000 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 112000 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 860160 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 40159 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2488320000 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 65633 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 9676800000 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 67 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 928972800 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 14 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 132710400 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 1933949 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 677376000000 </mn> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 23351 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 199065600000 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3096576000 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 27869184000 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> q </mi> <mn> 11 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> SeriesData </ci> <ci> $CellContext`q </ci> <cn type='integer'> 0 </cn> <list> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -3 <sep /> 10 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -17 <sep /> 900 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 120 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -1 <sep /> 800 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 29 <sep /> 48000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1 <sep /> 8064 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 10 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 18839 <sep /> 103680000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1 <sep /> 112000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 860160 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 40159 <sep /> 2488320000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -65633 <sep /> 9676800000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 67 <sep /> 928972800 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 10 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1 <sep /> 132710400 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 14 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -1933949 <sep /> 677376000000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 23351 <sep /> 199065600000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1 <sep /> 3096576000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 27869184000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> </list> <cn type='integer'> 0 </cn> <cn type='integer'> 11 </cn> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <ci> Se </ci> </apply> <apply> <ci> MathieuCharacteristicB </ci> <cn type='integer'> 4 </cn> <ci> q </ci> </apply> <ci> q </ci> <ci> z </ci> </apply> <apply> <ci> SeriesData </ci> <ci> $CellContext`q </ci> <cn type='integer'> 0 </cn> <list> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -3 <sep /> 10 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -17 <sep /> 900 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 120 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -1 <sep /> 800 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 29 <sep /> 48000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1 <sep /> 8064 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 10 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 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Date Added to functions.wolfram.com (modification date)
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