Derivative of the odd Mathieu function: Series representations (formula 11.04.06.0006)

MathieuSPrime

$Mathematica Notation$

Mathieu and Spheroidal Functions

MathieuSPrime[a,q,z]

Series representations

Generalized power series

Expansions at q==0

http://functions.wolfram.com/11.04.06.0006.01

Input Form

MathieuSPrime[MathieuCharacteristicB[3, q], q, z] \[Proportional] 3 Cos[3 z] + (Cos[z]/8 - (5/16) Cos[5 z]) q + (-(Cos[z]/64) - (15/512) Cos[3 z] + (7/640) Cos[7 z]) q^2 + (-(Cos[z]/4096) + (3/512) Cos[3 z] + (11 Cos[5 z])/8192 - Cos[9 z]/5120) q^3 + ((21 Cos[z])/32768 - (4863 Cos[3 z])/13107200 - (5 Cos[5 z])/16384 - (77 Cos[7 z])/2949120 + (11 Cos[11 z])/5160960) q^4 + (-((14061 Cos[z])/104857600) - (27 Cos[3 z])/131072 + (12329 Cos[5 z])/377487360 + (21 Cos[7 z])/3276800 + Cos[9 z]/3670016 - (13 Cos[13 z])/825753600) q^5 + (-((699 Cos[z])/838860800) + (13050583 Cos[3 z])/181193932800 + (533 Cos[5 z])/41943040 - (76679 Cos[7 z])/75497472000 - (17 Cos[9 z])/235929600 - (11 Cos[11 z])/6606028800 + Cos[15 z]/11890851840) q^6 + SeriesData[$CellContext`q, 0, {}, 0, 11, 1]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["MathieuSPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List["3", ",", "q"]], "]"]], ",", "q", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["3", " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Cos", "[", "z", "]"]], "8"], "-", RowBox[List[FractionBox["5", "16"], " ", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]]]]]], ")"]], " ", "q"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Cos", "[", "z", "]"]], "64"]]], "-", RowBox[List[FractionBox["15", "512"], " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List[FractionBox["7", "640"], " ", RowBox[List["Cos", "[", RowBox[List["7", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["q", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Cos", "[", "z", "]"]], "4096"]]], "+", RowBox[List[FractionBox["3", "512"], " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["11", " ", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]]]], "8192"], "-", FractionBox[RowBox[List["Cos", "[", RowBox[List["9", " ", "z"]], "]"]], "5120"]]], ")"]], " ", SuperscriptBox["q", "3"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["21", " ", RowBox[List["Cos", "[", "z", "]"]]]], "32768"], "-", FractionBox[RowBox[List["4863", " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "13107200"], "-", FractionBox[RowBox[List["5", " ", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]]]], "16384"], "-", FractionBox[RowBox[List["77", " ", RowBox[List["Cos", "[", RowBox[List["7", " ", "z"]], "]"]]]], "2949120"], "+", FractionBox[RowBox[List["11", " ", RowBox[List["Cos", "[", RowBox[List["11", " ", "z"]], "]"]]]], "5160960"]]], ")"]], " ", SuperscriptBox["q", "4"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["14061", " ", RowBox[List["Cos", "[", "z", "]"]]]], "104857600"]]], "-", FractionBox[RowBox[List["27", " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "131072"], "+", FractionBox[RowBox[List["12329", " ", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]]]], "377487360"], "+", FractionBox[RowBox[List["21", " ", RowBox[List["Cos", "[", RowBox[List["7", " ", "z"]], "]"]]]], "3276800"], "+", FractionBox[RowBox[List["Cos", "[", RowBox[List["9", " ", "z"]], "]"]], "3670016"], "-", FractionBox[RowBox[List["13", " ", RowBox[List["Cos", "[", RowBox[List["13", " ", "z"]], "]"]]]], "825753600"]]], ")"]], " ", SuperscriptBox["q", "5"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["699", " ", RowBox[List["Cos", "[", "z", "]"]]]], "838860800"]]], "+", FractionBox[RowBox[List["13050583", " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "181193932800"], "+", FractionBox[RowBox[List["533", " ", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]]]], "41943040"], "-", FractionBox[RowBox[List["76679", " ", RowBox[List["Cos", "[", RowBox[List["7", " ", "z"]], "]"]]]], "75497472000"], "-", FractionBox[RowBox[List["17", " ", RowBox[List["Cos", "[", RowBox[List["9", " ", "z"]], "]"]]]], "235929600"], "-", FractionBox[RowBox[List["11", " ", RowBox[List["Cos", "[", RowBox[List["11", " ", "z"]], "]"]]]], "6606028800"], "+", FractionBox[RowBox[List["Cos", "[", RowBox[List["15", " ", "z"]], "]"]], "11890851840"]]], ")"]], " ", SuperscriptBox["q", "6"]]], "+", InterpretationBox[SuperscriptBox[RowBox[List["O", "[", "q", "]"]], "11"], SeriesData[$CellContext`q, 0, List[], 0, 11, 1]]]]]]]]

MathML Form

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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'> 3 </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'> 3 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <cos /> <ci> $CellContext`z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> -5 <sep /> 16 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -1 <sep /> 64 </cn> <apply> <cos /> <ci> $CellContext`z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> -15 <sep /> 512 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 7 <sep /> 640 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -1 <sep /> 4096 </cn> <apply> <cos /> <ci> $CellContext`z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 3 <sep /> 512 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 11 <sep /> 8192 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1 <sep /> 5120 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 21 <sep /> 32768 </cn> <apply> <cos /> <ci> $CellContext`z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> -4863 <sep /> 13107200 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -5 <sep /> 16384 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -77 <sep /> 2949120 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 11 <sep /> 5160960 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> $CellContext`z </ci> </apply> </apply> 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type='integer'> 13 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> -699 <sep /> 838860800 </cn> <apply> <cos /> <ci> $CellContext`z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 13050583 <sep /> 181193932800 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 533 <sep /> 41943040 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -76679 <sep /> 75497472000 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -17 <sep /> 235929600 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -11 <sep /> 6606028800 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 11890851840 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 15 </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> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29