Derivative of the odd Mathieu function: Series representations (formula 11.04.06.0001)

MathieuSPrime

$Mathematica Notation$

Mathieu and Spheroidal Functions

MathieuSPrime[a,q,z]

Series representations

Generalized power series

Expansions at z==0

http://functions.wolfram.com/11.04.06.0001.01

Input Form

MathieuSPrime[MathieuCharacteristicB[2 n + 2, q], q, z] == 2 Sum[(k + 1) Subscript[B, 2 k + 2]^(2 n + 2) Cos[(2 k + 2) z], {k, 0, Infinity}] /; (MathieuCharacteristicB[2 n, q] - 4) Subscript[B, 2]^(2 n + 2) - q Subscript[B, 4]^(2 n + 2) == 0 && (MathieuCharacteristicB[2 n, q] - 4 k^2) Subscript[B, 2 k]^(2 n) - q (Subscript[B, 2 k + 2]^(2 n + 2) + Subscript[B, 2 k - 2]^(2 n + 2)) == 0 && Sum[Subscript[B, 2 k + 1]^(2 n + 1), {k, 0, Infinity}] == 1 && Element[n, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MathieuSPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "+", "2"]], ",", "q"]], "]"]], ",", "q", ",", "z"]], "]"]], "\[Equal]", RowBox[List["2", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], SubsuperscriptBox["B", RowBox[List[" ", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]]], RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "2"]], ")"]], " ", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List[RowBox[List["2", "n"]], ",", "q"]], "]"]], "-", "4"]], ")"]], SubsuperscriptBox["B", RowBox[List[" ", "2"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]]]]], "-", RowBox[List["q", " ", SubsuperscriptBox["B", RowBox[List[" ", "4"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]]]]]]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List[RowBox[List["2", "n"]], ",", "q"]], "]"]], "-", RowBox[List["4", SuperscriptBox["k", "2"]]]]], ")"]], SubsuperscriptBox["B", RowBox[List[" ", RowBox[List["2", "k"]]]], RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]]]]], "-", RowBox[List["q", RowBox[List["(", " ", RowBox[List[SubsuperscriptBox["B", RowBox[List[" ", RowBox[List[RowBox[List["2", "k"]], "+", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]]], "+", SubsuperscriptBox["B", RowBox[List[" ", RowBox[List[RowBox[List["2", "k"]], "-", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]]]]], ")"]]]]]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SubsuperscriptBox["B", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]], "\[Equal]", "1"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </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> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> B </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> b </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicB </ci> </annotation-xml> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> B </mi> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> - </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <msubsup> <mi> B </mi> <mn> 4 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> b </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicB </ci> </annotation-xml> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> B </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msubsup> </mrow> <mo> - </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> B </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> B </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <msubsup> <mi> B </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <ci> Se </ci> </apply> <apply> <ci> MathieuCharacteristicB </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <ci> q </ci> </apply> <ci> q </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ci> MathieuCharacteristicB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> q </ci> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> q </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ci> MathieuCharacteristicB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> q </ci> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> B </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MathieuSPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n_"]], "+", "2"]], ",", "q_"]], "]"]], ",", "q_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", SubsuperscriptBox["B", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], ")"]], " ", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", "q"]], "]"]], "-", "4"]], ")"]], " ", SubsuperscriptBox["B", "2", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]]]], "-", RowBox[List["q", " ", SubsuperscriptBox["B", "4", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]]]]]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicB", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", "q"]], "]"]], "-", RowBox[List["4", " ", SuperscriptBox["k", "2"]]]]], ")"]], " ", SubsuperscriptBox["B", RowBox[List["2", " ", "k"]], RowBox[List["2", " ", "n"]]]]], "-", RowBox[List["q", " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["B", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]], "+", SubsuperscriptBox["B", RowBox[List[RowBox[List["2", " ", "k"]], "-", "2"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]]]], ")"]]]]]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SubsuperscriptBox["B", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], "\[Equal]", "1"]], "&&", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]]]

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

2001-10-29