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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/03.01.13.0023.01









  


  










Input Form





z^2 Derivative[2][w][z] + (1 - 2 p) z Derivative[1][w][z] + (m^2 q^2 z^(2 q) + p^2 - \[Nu]^2 q^2) w[z] == 0 /; w[z] == Subscript[c, 1] z^p BesselJ[\[Nu], m z^q] + Subscript[c, 2] z^p BesselJ[-\[Nu], m z^q] && !Element[\[Nu], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "p"]]]], ")"]], " ", "z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], " ", SuperscriptBox["q", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "q"]]]]], "+", SuperscriptBox["p", "2"], "-", RowBox[List[SuperscriptBox["\[Nu]", "2"], " ", SuperscriptBox["q", "2"]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "p"], RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["m", " ", SuperscriptBox["z", "q"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "p"], RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["m", " ", SuperscriptBox["z", "q"]]]]], "]"]]]]]]]], "\[And]", " ", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]










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> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> q </mi> </mrow> </msup> </mrow> <mo> + </mo> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> p </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> q </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> p </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> q </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </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'> 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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> </apply> </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> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> </apply> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <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> p </ci> </apply> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <ci> q </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> p </ci> </apply> <apply> <ci> BesselJ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> &#957; </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "p_"]]]], ")"]], " ", "z_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["m_", "2"], " ", SuperscriptBox["q_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "q_"]]]]], "+", SuperscriptBox["p_", "2"], "-", RowBox[List[SuperscriptBox["\[Nu]_", "2"], " ", SuperscriptBox["q_", "2"]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "p"], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["m", " ", SuperscriptBox["z", "q"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "p"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["m", " ", SuperscriptBox["z", "q"]]]]], "]"]]]]]]]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]










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





2001-10-29





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