html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 BesselJ

 http://functions.wolfram.com/03.01.22.0010.01

 Input Form

 LaplaceTransform[Subscript[j, n][t], t, z] == z^n (ArcTan[1/z] Sum[Subscript[c, k, l]/z^(2 k), {k, 0, Floor[n/2]}] - Sum[Subscript[c, k, l] Sum[((-1)^m/(2 m + 1)) z^(-2 k - 2 m - 1), {m, 0, Floor[(n - 1)/2] - k}], {k, 0, Floor[n/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["LaplaceTransform", "[", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "t", "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", FractionBox["1", "z"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "l"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "l"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]], "-", "k"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " "]], RowBox[List[RowBox[List["2", " ", "m"]], "+", "1"]]], SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", RowBox[List["2", " ", "m"]], "-", "1"]]]]]]]]]]]]], ")"]]]]]], "/;", 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

 t [ j n ( t ) ] ( z ) z n ( tan - 1 ( 1 z ) k = 0 n 2 c k , l z - 2 k - k = 0 n 2 c k , l m = 0 n - 1 2 - k ( - 1 ) m 2 m + 1 z - 2 k - 2 m - 1 ) /; n + j n ( z ) π 2 z J n + 1 2 ( z ) c 0 , 0 0 c 1 , 2 1 2 c 0 , n ( - 1 ) n ( 2 n - 1 ) !! n ! ( c k , n ( n - 1 ) c k - 1 , n - 2 n - ( 2 n - 1 ) c k , n - 1 n /; n 1 k n ) ( c k , n 0 /; k > n ) Condition LaplaceTransform SphericalBesselJ n t t z z n 1 z -1 k 0 n 2 -1 Subscript c k l z -2 k -1 k 0 n 2 -1 Subscript c k l m 0 n -1 2 -1 -1 k -1 m 2 m 1 -1 z -2 k -1 2 m -1 n SuperPlus SphericalBesselJ n z 2 z -1 1 2 BesselJ n 1 2 z Subscript c 0 0 0 Subscript c 1 2 1 2 Subscript c 0 n -1 n Factorial2 2 n -1 n -1 Condition Subscript c k n n -1 Subscript c k -1 n -2 n -1 -1 2 n -1 Subscript c k n -1 n -1 n 1 k n Condition Subscript c k n 0 k n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "t_", "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", FractionBox["1", "z"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "l"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "l"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]], "-", "k"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", RowBox[List["2", " ", "m"]], "-", "1"]]]]], RowBox[List[RowBox[List["2", " ", "m"]], "+", "1"]]]]]]]]]]], ")"]]]], "/;", 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