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SpheroidalPSPrime




Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalPSPrime[nu,mu,gamma,z] > Specific values > Specialized values > For fixed nu, gamma, z





http://functions.wolfram.com/11.12.03.0002.01









  


  








Input Form





SpheroidalPSPrime[\[Nu], 1/2, \[Gamma], z] == (1/(Sqrt[2 Pi] (1 - z^2)^(5/4))) (z MathieuC[MathieuCharacteristicA[1/2 + \[Nu], \[Gamma]^2/4], \[Gamma]^2/4, ArcCos[z]] - 2 Sqrt[1 - z^2] MathieuCPrime[MathieuCharacteristicA[1/2 + \[Nu], \[Gamma]^2/4], \[Gamma]^2/4, ArcCos[z]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]", ",", FractionBox["1", "2"], ",", "\[Gamma]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["5", "/", "4"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["MathieuC", "[", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"]]], "]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"], ",", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["MathieuCPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"]]], "]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"], ",", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]]]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <msup> <msub> <mi> PS </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </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[FractionBox[&quot;1&quot;, &quot;2&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> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> Ce </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> a </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicA </ci> </annotation-xml> </semantics> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mfrac> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> , </mo> <mfrac> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <mi> Ce </mi> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> a </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicA </ci> </annotation-xml> </semantics> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mfrac> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> , </mo> <mfrac> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SpheroidalPSPrime </ci> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> &#947; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 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='rational'> 5 <sep /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <ci> Ce </ci> <apply> <ci> MathieuCharacteristicA </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <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='rational'> 1 <sep /> 2 </cn> </apply> <apply> <apply> <ci> Subscript </ci> <ci> Ce </ci> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <ci> MathieuCharacteristicA </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]_", ",", FractionBox["1", "2"], ",", "\[Gamma]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["z", " ", RowBox[List["MathieuC", "[", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"]]], "]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"], ",", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["MathieuCPrime", "[", RowBox[List[RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"]]], "]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"], ",", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]]]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["5", "/", "4"]]]]]]]]]]


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





2007-05-02