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SpheroidalPS






Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalPS[nu,mu,gamma,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/11.08.21.0002.01









  


  










Input Form





Integrate[SpheroidalPS[Subscript[n, 1], m, \[Gamma], t] SpheroidalPS[Subscript[n, 2], m, \[Gamma], t], {t, -1, 1}] == ((2 (m + Subscript[n, 1])!)/((2 Subscript[n, 1] + 1) (Subscript[n, 1] - m)!)) KroneckerDelta[Subscript[n, 1], Subscript[n, 2]] /; Element[Subscript[n, 1], Integers] && Element[Subscript[n, 2], Integers] && Element[m, Integers] && Subscript[n, 1] >= m && Subscript[n, 2] >= m










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> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <semantics> <mrow> <msub> <mi> PS </mi> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mi> m </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox[&quot;PS&quot;, &quot;IT&quot;], RowBox[List[TagBox[SubscriptBox[&quot;n&quot;, &quot;1&quot;], SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;m&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;t&quot;, SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalPS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> PS </mi> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mi> m </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox[&quot;PS&quot;, &quot;IT&quot;], RowBox[List[TagBox[SubscriptBox[&quot;n&quot;, &quot;2&quot;], SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;m&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;t&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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </msub> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8805; </mo> <mi> m </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8805; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <ci> SpheroidalPS </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> <ci> &#947; </ci> <ci> t </ci> </apply> <apply> <ci> SpheroidalPS </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> <ci> &#947; </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <geq /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <apply> <geq /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> </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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