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SpheroidalPSPrime






Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalPSPrime[nu,mu,gamma,z] > Differentiation > Low-order differentiation > With respect to z





http://functions.wolfram.com/11.12.20.0002.01









  


  










Input Form





D[SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], z], {z, 2}] == (1/(-1 + z^2)^3) (2 z ((-1 + z^2)^2 \[Gamma]^2 - 3 \[Mu]^2 - 2 (-1 + z^2) SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]) SpheroidalPS[\[Nu], \[Mu], \[Gamma], z] - (-1 + z^2) (-2 + \[Gamma]^2 + z^4 \[Gamma]^2 - 2 z^2 (3 + \[Gamma]^2) - \[Mu]^2 - (-1 + z^2) SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]) SpheroidalPSPrime[\[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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <semantics> <mrow> <msup> <msub> <mi> PS </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mi> z </mi> </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;z&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> &#8706; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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</mo> <semantics> <mrow> <msup> <msub> <mi> PS </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mi> z </mi> </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;z&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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> SpheroidalPSPrime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> </apply> </apply> </apply> </apply> <apply> <ci> SpheroidalPS </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <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> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <ci> SpheroidalPSPrime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <ci> z </ci> </apply> </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





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