Derivative of the odd Mathieu function: Series representations (formula 11.04.06.0004)

MathieuSPrime

$Mathematica Notation$

Mathieu and Spheroidal Functions

MathieuSPrime[a,q,z]

Series representations

Generalized power series

Expansions at q==0

http://functions.wolfram.com/11.04.06.0004.01

Input Form

MathieuSPrime[MathieuCharacteristicB[1, q], q, z] \[Proportional] Cos[z] - (3/8) Cos[3 z] q + (-(Cos[z]/128) + (3/64) Cos[3 z] + (5/192) Cos[5 z]) q^2 + (Cos[z]/512 + Cos[3 z]/1024 - (5 Cos[5 z])/1152 - (7 Cos[7 z])/9216) q^3 + (-((37 Cos[z])/294912) - (49 Cos[3 z])/24576 + (7 Cos[7 z])/49152 + Cos[9 z]/81920) q^4 + (-((121 Cos[z])/1769472) + (317 Cos[3 z])/786432 + (205 Cos[5 z])/1179648 - (7 Cos[7 z])/5898240 - Cos[9 z]/409600 - (11 Cos[11 z])/88473600) q^5 + ((8105 Cos[z])/339738624 + (103 Cos[3 z])/18874368 - (731 Cos[5 z])/18874368 - (2653 Cos[7 z])/471859200 + Cos[9 z]/31457280 + (11 Cos[11 z])/424673280 + (13 Cos[13 z])/14863564800) q^6 + O[q]^11

Standard Form

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MathML Form

<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> 1 </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> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 128 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 64 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 192 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 5 </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> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 512 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 1024 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> 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<mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 409600 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 5898240 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 88473600 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 8105 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 339738624 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 103 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 18874368 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 31457280 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 424673280 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 13 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 13 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 14863564800 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 731 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 18874368 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 2653 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 471859200 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <semantics> <msup> <mrow> <mi> O </mi> <mo> [ </mo> <mi> q </mi> <mo> ] </mo> </mrow> <mn> 11 </mn> </msup> <annotation-xml encoding='MathML-Content'> <apply> <ci> SeriesData </ci> <ci> $CellContext`q </ci> <cn type='integer'> 0 </cn> <list /> <cn type='integer'> 0 </cn> <cn type='integer'> 11 </cn> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> </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'> 1 </cn> <ci> q </ci> </apply> <ci> q </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <cos /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 3 <sep /> 8 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <ci> q </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 128 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 3 <sep /> 64 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 5 <sep /> 192 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 512 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 1024 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 1152 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 9216 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 37 </cn> <apply> <cos /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 294912 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 49152 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 81920 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 49 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 24576 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 121 </cn> <apply> <cos /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 1769472 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 317 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 786432 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 205 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 1179648 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 409600 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103 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 18874368 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 31457280 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 424673280 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 13 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 14863564800 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 731 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 18874368 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2653 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 471859200 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <ci> SeriesData </ci> <ci> $CellContext`q </ci> <cn type='integer'> 0 </cn> <list /> <cn type='integer'> 0 </cn> <cn type='integer'> 11 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29