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http://functions.wolfram.com/03.15.13.0007.01
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z^4 Derivative[4][w][z] + (6 - 4 r - 4 s) z^3 Derivative[3][w][z] +
z^2 (7 + 12 r (-1 + s) + 6 (-2 + s) s + 4 r^2) Derivative[2][w][z] +
(-1 + 2 r + 2 s) z (-1 + r (2 - 4 s) - 2 (-1 + s) s) Derivative[1][w][z] +
(4 r s^3 + s^4 + 4 r^2 s^2 + a^4 r^4 z^(4 r)) w[z] == 0 /;
w[z] == Subscript[c, 1] z^s KelvinBer[a z^r] +
Subscript[c, 2] z^s KelvinBei[a z^r] + Subscript[c, 3] z^s
KelvinKer[a z^r] + Subscript[c, 4] z^s KelvinKei[a z^r]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "4", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["6", "-", RowBox[List["4", " ", "r"]], "-", RowBox[List["4", " ", "s"]]]], ")"]], SuperscriptBox["z", "3"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["12", " ", "r", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "s"]], ")"]]]], "+", RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "s"]], ")"]], " ", "s"]], "+", RowBox[List["4", " ", SuperscriptBox["r", "2"]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "r"]], "+", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["4", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "s"]], ")"]], " ", "s"]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "r", " ", SuperscriptBox["s", "3"]]], "+", SuperscriptBox["s", "4"], "+", RowBox[List["4", " ", SuperscriptBox["r", "2"], " ", SuperscriptBox["s", "2"]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["r", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]]]], ")"]], " ", 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["KelvinBer", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["z", "s"], RowBox[List["KelvinBei", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], SuperscriptBox["z", "s"], RowBox[List["KelvinKer", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "4"], SuperscriptBox["z", "s"], RowBox[List["KelvinKei", "[", 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> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <semantics> <mrow> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "4", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "3", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> <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> a </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> + </mo> <msup> <mi> s </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <msup> <mi> s </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> <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> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> ber </mi> <mo> ⁡ </mo> <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> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> bei </mi> <mo> ⁡ </mo> <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> 3 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> ker </mi> <mo> ⁡ </mo> <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> 4 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> kei </mi> <mo> ⁡ </mo> <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 /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 4 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <ci> s </ci> <cn type='integer'> -2 </cn> </apply> <ci> s </ci> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <ci> s </ci> </apply> <apply> <times /> <ci> r </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> <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> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <power /> <ci> s </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> <apply> <power /> <ci> s </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> <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> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinBer </ci> <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> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinBei </ci> <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'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinKer </ci> <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'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinKei </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </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[SuperscriptBox["z_", "4"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "4", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["6", "-", RowBox[List["4", " ", "r_"]], "-", RowBox[List["4", " ", "s_"]]]], ")"]], " ", SuperscriptBox["z_", "3"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["12", " ", "r_", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "s_"]], ")"]]]], "+", RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "s_"]], ")"]], " ", "s_"]], "+", RowBox[List["4", " ", SuperscriptBox["r_", "2"]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "r_"]], "+", RowBox[List["2", " ", "s_"]]]], ")"]], " ", "z_", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["r_", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["4", " ", "s_"]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "s_"]], ")"]], " ", "s_"]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "r_", " ", SuperscriptBox["s_", "3"]]], "+", SuperscriptBox["s_", "4"], "+", RowBox[List["4", " ", SuperscriptBox["r_", "2"], " ", SuperscriptBox["s_", "2"]]], "+", RowBox[List[SuperscriptBox["a_", "4"], " ", SuperscriptBox["r_", "4"], " ", SuperscriptBox["z_", RowBox[List["4", " ", "r_"]]]]]]], ")"]], " ", 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["KelvinBer", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["KelvinBei", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["KelvinKer", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "4"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["KelvinKei", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
