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SpheroidalPS




Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalPS[nu,mu,gamma,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/11.08.13.0001.01









  


  








Input Form





(1 - z^2) Derivative[2][w][z] - 2 z Derivative[1][w][z] + (SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] + \[Gamma]^2 (1 - z^2) - \[Mu]^2/(1 - z^2)) w[z] == 0 /; w[z] == Subscript[c, 1] SpheroidalPS[\[Nu], \[Mu], \[Gamma], z] + Subscript[c, 2] SpheroidalQS[\[Nu], \[Mu], \[Gamma], z]








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> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mfrac> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> PS </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox[&quot;PS&quot;, &quot;IT&quot;], RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;z&quot;, SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalPS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> QS </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox[&quot;QS&quot;, &quot;IT&quot;], RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;z&quot;, SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mtext> </mtext> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> SpheroidalPS </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> SpheroidalQS </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02