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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Integral representations > Integral representations of negative integer order





http://functions.wolfram.com/03.01.07.0013.01









  


  










Input Form





Subscript[j, n][z] == Sum[Subscript[c, k, n] D[Subscript[j, 0][z], {x, n - 2 k}], {k, 0, Floor[l/2]}] /; Element[n, Integers] && n >= 1 && Subscript[j, n][z] == Sqrt[Pi/(2 z)] BesselJ[n + 1/2, z] && Subscript[c, 0, 0] == 0 && Subscript[c, 1, 2] == 1/2 && Subscript[c, 0, n] == (-1)^n ((2 n - 1)!!/n!) && (Subscript[c, k, n] == (-((2 n - 1)/n)) Subscript[c, k, n - 1] + ((n - 1)/n) Subscript[c, k - 1, n - 2] /; n >= 1 && k <= n) && (Subscript[c, k, n] == 0 /; k > n)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["l", "2"], "]"]]], RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "n"]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["x", ",", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]]]], "}"]]], RowBox[List[SubscriptBox["j", "0"], "[", "z", "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", RowBox[List["2", " ", "z"]]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]], "\[And]", RowBox[List[SubscriptBox["c", RowBox[List["0", ",", "0"]]], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["c", RowBox[List["1", ",", "2"]]], "\[Equal]", FractionBox["1", "2"]]], "\[And]", RowBox[List[SubscriptBox["c", RowBox[List["0", ",", "n"]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], "!!"]], " "]], RowBox[List["n", "!"]]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "n"]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1", " "]], "n"]]], SubscriptBox["c", RowBox[List["k", ",", RowBox[List["n", "-", "1"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List["n", "-", "1", " "]], "n"], SubscriptBox["c", RowBox[List[RowBox[List["k", "-", "1"]], ",", RowBox[List["n", "-", "2"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List["k", "\[LessEqual]", "n"]]]]]], ")"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "n"]]], "\[Equal]", "0"]], "/;", RowBox[List["k", ">", "n"]]]], ")"]]]]]]]]










MathML Form







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</mo> <msub> <mi> c </mi> <mrow> <mi> k </mi> <mo> , </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </msub> </mrow> <mi> n </mi> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> c </mi> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &gt; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> l </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </degree> </bvar> <apply> <ci> SphericalBesselJ </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <geq /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <ci> k </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> <ci> n </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <ci> k </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["j", "n"], "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["l", "2"], "]"]]], RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "n"]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["x", ",", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]]]], "}"]]]]], RowBox[List[SubscriptBox["j", "0"], "[", "z", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", RowBox[List["2", " ", "z"]]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]], "&&", RowBox[List[SubscriptBox["c", RowBox[List["0", ",", "0"]]], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["c", RowBox[List["1", ",", "2"]]], "\[Equal]", FractionBox["1", "2"]]], "&&", RowBox[List[SubscriptBox["c", RowBox[List["0", ",", "n"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], "!!"]]]], RowBox[List["n", "!"]]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "n"]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], " ", SubscriptBox["c", RowBox[List["k", ",", RowBox[List["n", "-", "1"]]]]]]], "n"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SubscriptBox["c", RowBox[List[RowBox[List["k", "-", "1"]], ",", RowBox[List["n", "-", "2"]]]]]]], "n"]]]]], "/;", RowBox[List[RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List["k", "\[LessEqual]", "n"]]]]]], ")"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "n"]]], "\[Equal]", "0"]], "/;", RowBox[List["k", ">", "n"]]]], ")"]]]]]]]]]]










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





2002-12-18





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