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; }

 SphericalBesselJ

 http://functions.wolfram.com/03.21.20.0018.01

 Input Form

 D[SphericalBesselJ[\[Nu], z], {z, n}] == Sum[Binomial[n, i] Pochhammer[1/2 - n + i, n - i] Sum[(-1)^(i + m) Binomial[i, m] Pochhammer[-(1/2) - \[Nu], i - m] Sum[(((-1)^(-1 + k) 2^(2 k - m) Pochhammer[-m, 2 (-k + m)] Pochhammer[1/2 + \[Nu], k])/(-k + m)!) (-Sum[(((-j + k)! z^(2 j))/(4^j (j! (-2 j + k)! Pochhammer[ 1/2 - k - \[Nu], j] Pochhammer[1/2 + \[Nu], j]))) SphericalBesselJ[\[Nu], z], {j, 0, k}] + (1/2) z Sum[(((-1 - j + k)! z^(2 j))/(4^j (j! (-1 - 2 j + k)! Pochhammer[1/2 - k - \[Nu], j] Pochhammer[1/2 + \[Nu], 1 + j]))) SphericalBesselJ[\[Nu] - 1, z], {j, 0, -1 + k}]), {k, 0, m}], {m, 0, i}], {i, 0, n}]/z^n /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", " ", RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n", "+", "i"]], ",", RowBox[List["n", "-", "i"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "+", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["i", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], ",", RowBox[List["i", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "j"]]]]], ")"]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "j"]], "]"]]]]], RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", "k"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "j"]]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "j"]]]], "]"]]]]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n j ν ( z ) z n z - n i = 0 n ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["i", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( i - n + 1 2 ) n - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["i", "-", "n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["n", "-", "i"]]], Pochhammer] m = 0 i ( - 1 ) i + m ( i m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["m", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - ν - 1 2 ) i - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], ")"]], RowBox[List["i", "-", "m"]]], Pochhammer] k = 0 m ( - 1 ) k - 1 2 2 k - m ( - m ) 2 ( m - k ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]]]]], Pochhammer] ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( m - k ) ! ( 1 2 z j = 0 k - 1 ( - j + k - 1 ) ! 4 - j z 2 j j ! ( - 2 j + k - 1 ) ! ( - k - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( ν + 1 2 ) j + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] j ν - 1 ( z ) - j = 0 k ( k - j ) ! 4 - j z 2 j j ! ( k - 2 j ) ! ( - k - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] j ν ( z ) ) /; n Condition z n SphericalBesselJ ν z z -1 n i 0 n Binomial n i Pochhammer i -1 n 1 2 n -1 i m 0 i -1 i m Binomial i m Pochhammer -1 ν -1 1 2 i -1 m k 0 m -1 k -1 2 2 k -1 m Pochhammer -1 m 2 m -1 k Pochhammer ν 1 2 k m -1 k -1 1 2 z j 0 k -1 -1 j k -1 4 -1 j z 2 j j -2 j k -1 Pochhammer -1 k -1 ν 1 2 j Pochhammer ν 1 2 j 1 -1 SphericalBesselJ ν -1 z -1 j 0 k k -1 j 4 -1 j z 2 j j k -1 2 j Pochhammer -1 k -1 ν 1 2 j Pochhammer ν 1 2 j -1 SphericalBesselJ ν z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n", "+", "i"]], ",", RowBox[List["n", "-", "i"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "+", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["i", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]], ",", RowBox[List["i", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "k"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "j"]]]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "j"]], "]"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", "k"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "j"]]]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "j"]], "+", "k"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "j"]]]], "]"]]]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ")"]], "!"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02

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