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 SphericalBesselJ

 http://functions.wolfram.com/03.21.06.0015.01

 Input Form

 SphericalBesselJ[\[Nu], z] == Exp[2 \[Nu] Pi I Floor[Arg[z - x]/(2 Pi)]] Sum[(1/(x^k k!)) Sum[Binomial[k, i] Pochhammer[1/2 - k + i, k - i] Sum[(-1)^(i + m) Binomial[i, m] Pochhammer[-(1/2) - \[Nu], i - m] Sum[(((-1)^(-1 + u) 2^(2 u - m) Pochhammer[-m, 2 (-u + m)] Pochhammer[1/2 + \[Nu], u])/(-u + m)!) (-Sum[(((-j + u)! x^(2 j))/(4^j (j! (-2 j + u)! Pochhammer[ 1/2 - u - \[Nu], j] Pochhammer[1/2 + \[Nu], j]))) SphericalBesselJ[\[Nu], x], {j, 0, u}] + (1/2) x Sum[(((-1 - j + u)! x^(2 j))/(4^j (j! (-1 - 2 j + u)! Pochhammer[1/2 - u - \[Nu], j] Pochhammer[1/2 + \[Nu], 1 + j]))) SphericalBesselJ[\[Nu] - 1, x], {j, 0, -1 + u}]) (z - x)^k, {u, 0, m}], {m, 0, i}], {i, 0, k}], {k, 0, Infinity}] /; Element[x, Reals] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["Exp", "[", RowBox[List["2", "\[Nu]", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["x", RowBox[List["-", "k"]]], RowBox[List["k", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k", "+", "i"]], ",", RowBox[List["k", "-", "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["u", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "u"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "u"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "u"]], "+", "m"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "u"]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "u"]], "+", "m"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "u"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "u"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["x", RowBox[List["2", "j"]]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "u"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "u", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "j"]], "]"]]]]], RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "x", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "u"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", "u"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["x", RowBox[List["2", "j"]]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "j"]], "+", "u"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "u", "-", "\[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"]], ",", "x"]], "]"]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 j ν ( z ) 2 ν π arg ( z - x ) 2 π k = 0 x - k k ! i = 0 k ( k i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", 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 - k + 1 2 ) k - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["i", "-", "k", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "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] u = 0 m ( - 1 ) u - 1 2 2 u - m ( - m ) 2 ( m - u ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "u"]], ")"]]]]], Pochhammer] ( ν + 1 2 ) u TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "u"], Pochhammer] ( m - u ) ! ( 1 2 x j = 0 u - 1 ( ( - j + u - 1 ) ! 4 - j x 2 j ) j ν - 1 ( x ) j ! ( - 2 j + u - 1 ) ! ( - u - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "u"]], "-", "\[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 = 0 u ( ( u - j ) ! 4 - j x 2 j ) j ν ( x ) j ! ( u - 2 j ) ! ( - u - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "u"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition SphericalBesselJ ν z 2 ν z -1 x 2 -1 k 0 x -1 k k -1 i 0 k Binomial k i Pochhammer i -1 k 1 2 k -1 i m 0 i -1 i m Binomial i m Pochhammer -1 ν -1 1 2 i -1 m u 0 m -1 u -1 2 2 u -1 m Pochhammer -1 m 2 m -1 u Pochhammer ν 1 2 u m -1 u -1 1 2 x j 0 u -1 -1 j u -1 4 -1 j x 2 j SphericalBesselJ ν -1 x j -2 j u -1 Pochhammer -1 u -1 ν 1 2 j Pochhammer ν 1 2 j 1 -1 -1 j 0 u u -1 j 4 -1 j x 2 j SphericalBesselJ ν x j u -1 2 j Pochhammer -1 u -1 ν 1 2 j Pochhammer ν 1 2 j -1 z -1 x k x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Nu]", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["x", RowBox[List["-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k", "+", "i"]], ",", RowBox[List["k", "-", "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["u", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "u"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "u"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "u"]], "+", "m"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "u"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "u"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "u"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["x", RowBox[List["2", " ", "j"]]]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "+", "u"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "u", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "j"]], "]"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "x", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "u"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", "u"]], ")"]], "!"]], " ", SuperscriptBox["4", RowBox[List["-", "j"]]], " ", SuperscriptBox["x", RowBox[List["2", " ", "j"]]]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "j"]], "+", "u"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "u", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "j"]]]], "]"]]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "u"]], "+", "m"]], ")"]], "!"]]]]]]]]]]]]]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2007-05-02