Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











MathieuCPrime






Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > MathieuCPrime[a,q,z] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/11.03.06.0002.01









  


  










Input Form





MathieuCPrime[MathieuCharacteristicA[2 n + 1, q], q, z] == -Sum[(2 k + 1) Subscript[A, 2 k + 1]^(2 n + 1) Sin[(2 k + 1) z], {k, 0, Infinity}] /; (MathieuCharacteristicA[2 n + 1, q] - 1 - q) Subscript[A, 1]^(2 n + 1) - q Subscript[A, 3]^(2 n + 1) == 0 && (MathieuCharacteristicA[2 n + 1, q] - (2 k + 1)^2) Subscript[A, 2 k + 1]^(2 n + 1) - q (Subscript[A, 2 k + 3]^(2 n + 1) + Subscript[A, 2 k - 1]^(2 n + 1)) == 0 && Sum[Subscript[A, 2 k + 1]^(2 n + 1), {k, 0, Infinity}] == 1 && Element[n, Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MathieuCPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ",", "q"]], "]"]], ",", "q", ",", "z"]], "]"]], "\[Equal]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], SubsuperscriptBox["A", RowBox[List[" ", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], " ", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ",", "q"]], "]"]], "-", "1", "-", "q"]], ")"]], SubsuperscriptBox["A", RowBox[List[" ", "1"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]], "-", RowBox[List["q", " ", SubsuperscriptBox["A", RowBox[List[" ", "3"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]]]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ",", "q"]], "]"]], "-", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], " ", "+", " ", "1"]], ")"]], "2"]]], ")"]], SubsuperscriptBox["A", RowBox[List[" ", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]], "-", RowBox[List["q", RowBox[List["(", " ", RowBox[List[SubsuperscriptBox["A", RowBox[List[" ", RowBox[List[RowBox[List["2", "k"]], "+", "3"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]], "+", SubsuperscriptBox["A", RowBox[List[" ", RowBox[List[RowBox[List["2", "k"]], "-", "1"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]], ")"]]]]]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SubsuperscriptBox["A", 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> Ce </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> a </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicA </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </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> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> A </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> a </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicA </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> A </mi> <mn> 1 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> - </mo> <mrow> <mi> q </mi> <mo> &#8290; </mo> <msubsup> <mi> A </mi> <mn> 3 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> a </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicA </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> A </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> - </mo> <mrow> <mi> q </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> A </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> A </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <msubsup> <mi> A </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, 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> Ce </ci> </apply> <apply> <ci> MathieuCharacteristicA </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> q </ci> </apply> <ci> q </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> A </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> <sin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ci> MathieuCharacteristicA </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> A </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> q </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> A </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ci> MathieuCharacteristicA </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> A </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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> q </ci> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> A </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> <power /> <apply> <ci> Subscript </ci> <ci> A </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </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> A </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["MathieuCPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]], ",", "q_"]], "]"]], ",", "q_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", SubsuperscriptBox["A", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ",", "q"]], "]"]], "-", "1", "-", "q"]], ")"]], " ", SubsuperscriptBox["A", "1", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], "-", RowBox[List["q", " ", SubsuperscriptBox["A", "3", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]]]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ",", "q"]], "]"]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"]]], ")"]], " ", SubsuperscriptBox["A", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], "-", RowBox[List["q", " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["A", RowBox[List[RowBox[List["2", " ", "k"]], "+", "3"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], "+", SubsuperscriptBox["A", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]]]], ")"]]]]]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], SubsuperscriptBox["A", 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





© 1998- Wolfram Research, Inc.