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http://functions.wolfram.com/07.45.13.0009.01
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4 z^2 Derivative[2][w][z] - 4 (-1 + r + 2 s) z Derivative[1][w][z] +
(4 r s + 4 s^2 + r^2 (1 - a^2 z^(2 r) - 4 \[Mu]^2 + 4 a z^r \[Nu]))
w[z] == 0 /; w[z] == Subscript[c, 1] z^s WhittakerW[\[Nu], \[Mu],
a z^r] + Subscript[c, 2] z^s WhittakerM[\[Nu], \[Mu], a z^r]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["4", " ", SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "r", "+", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "r", " ", "s"]], "+", RowBox[List["4", " ", SuperscriptBox["s", "2"]]], "+", RowBox[List[SuperscriptBox["r", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], "-", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["4", " ", "a", " ", SuperscriptBox["z", "r"], " ", "\[Nu]"]]]], ")"]]]]]], ")"]], " ", 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["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", RowBox[List["a", " ", 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> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> s </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> s </mi> <mn> 2 </mn> </msup> </mrow> </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> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mrow> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mi> W </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox["z", "s"], "W"]], WhittakerM] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msub> <semantics> <mrow> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mi> M </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox["z", "s"], "M"]], WhittakerW] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> <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 /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> ν </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> s </ci> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> s </ci> <cn type='integer'> 2 </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> <apply> <ci> Subscript </ci> <apply> <ci> WhittakerM </ci> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <ci> W </ci> </apply> </apply> <ci> ν </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </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> <ci> Subscript </ci> <apply> <ci> WhittakerW </ci> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <ci> M </ci> </apply> </apply> <ci> ν </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["4", " ", SuperscriptBox["z_", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "r_", "+", RowBox[List["2", " ", "s_"]]]], ")"]], " ", "z_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "r_", " ", "s_"]], "+", RowBox[List["4", " ", SuperscriptBox["s_", "2"]]], "+", RowBox[List[SuperscriptBox["r_", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]], "-", RowBox[List["4", " ", SuperscriptBox["\[Mu]_", "2"]]], "+", RowBox[List["4", " ", "a_", " ", SuperscriptBox["z_", "r_"], " ", "\[Nu]_"]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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