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http://functions.wolfram.com/11.01.13.0007.01
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Derivative[2][w][z] + ((1 - r - 2 s) Derivative[1][w][z])/z +
(b^2 r^2 z^(-2 + 2 r) (a - 2 q Cos[2 b z^r]) + (s (r + s))/z^2) w[z] ==
0 /; w[z] == Subscript[c, 1] z^s MathieuC[a, q, b z^r] +
Subscript[c, 2] z^s MathieuS[a, q, b z^r]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "r", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "1"]]], RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["r", "2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "r"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "q", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "b", " ", SuperscriptBox["z", "r"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["s", " ", RowBox[List["(", RowBox[List["r", "+", "s"]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "2"]]]]]]], ")"]], RowBox[List["w", "[", "z", "]"]]]]]], " ", "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["z", "s"], " ", RowBox[List["MathieuC", "[", RowBox[List["a", ",", "q", ",", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["z", "s"], " ", RowBox[List["MathieuS", "[", RowBox[List["a", ",", "q", ",", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> w </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> r </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mtext> </mtext> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Ce </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> q </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Se </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> q </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <times /> <ci> s </ci> <apply> <plus /> <ci> r </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> Ce </ci> <ci> a </ci> <ci> q </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> Se </ci> <ci> a </ci> <ci> q </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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