Derivative[0, n][BesselJ][\[Nu], 0] == 0 /; Element[n, Integers] && n > 0 && ((Element[\[Nu], Integers] && Abs[\[Nu]] > n) || (Element[(n - \[Nu] - 1)/2, Integers] && (n - \[Nu] - 1)/2 >= 0))

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["0", ",", "n"]], "]"]], "[", "BesselJ", "]"]], "[", RowBox[List["\[Nu]", ",", "0"]], "]"]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], ">", "n"]]]], "\[Or]", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "-", "\[Nu]", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "\[Nu]", "-", "1"]], "2"], "\[GreaterEqual]", "0"]]]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <semantics> <mi> J </mi> <annotation encoding='Mathematica'> TagBox[&quot;J&quot;, BesselJ] </annotation> </semantics> <mi> &#957; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;0&quot;, &quot;,&quot;, &quot;n&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msubsup> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mi> n </mi> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <ci> n </ci> </list> <apply> <ci> Subscript </ci> <apply> <ci> BesselJ </ci> <ci> J </ci> </apply> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <ci> &#957; </ci> <integers /> </apply> <apply> <gt /> <apply> <abs /> <ci> &#957; </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["BesselJ", TagBox[RowBox[List["(", RowBox[List["0", ",", "n"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "0"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], ">", "n"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "\[Nu]", "-", "1"]], ")"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "\[Nu]", "-", "1"]], ")"]]]], "\[GreaterEqual]", "0"]]]], ")"]]]], ")"]]]]]]]]]]

2001-10-29