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SpheroidalPS






Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalPS[nu,mu,gamma,z] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/11.08.06.0005.01









  


  










Input Form





SpheroidalPS[\[Nu], \[Mu], \[Gamma], z] \[Proportional] SpheroidalPS[\[Nu], \[Mu], \[Gamma], 0] + z SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], 0] - (1/2) (\[Gamma]^2 - \[Mu]^2 + SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]) SpheroidalPS[\[Nu], \[Mu], \[Gamma], 0] z^2 + (1/6) (2 SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], 0] - (\[Gamma]^2 - \[Mu]^2 + SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]) SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], 0]) z^3 + \[Ellipsis] /; (z -> 0)










Standard Form





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MathML Form







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</mi> </mrow> </msub> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubscriptBox[StyleBox[&quot;PS&quot;, &quot;IT&quot;], RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;\[Prime]&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;0&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalPSPrime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; 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</mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubscriptBox[StyleBox[&quot;PS&quot;, &quot;IT&quot;], RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;\[Prime]&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;0&quot;, SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalPSPrime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SpheroidalPS </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> SpheroidalPS </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <apply> <ci> SpheroidalPSPrime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <cn type='integer'> 0 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> </apply> </apply> <apply> <ci> SpheroidalPS </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> SpheroidalPSPrime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> </apply> </apply> <apply> <ci> SpheroidalPSPrime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "\[Gamma]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]], "+", RowBox[List["z", " ", RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Gamma]", "2"], "-", SuperscriptBox["\[Mu]", "2"], "+", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]]]], ")"]], " ", RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Gamma]", "2"], "-", SuperscriptBox["\[Mu]", "2"], "+", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]]]], ")"]], " ", RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", "3"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]










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





2007-05-02