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/03.21.03.0037.01

 Input Form

 SphericalBesselJ[\[Nu], z] == ((I^((-(1/3) + Abs[1/2 + \[Nu]]) (1 + Sign[1/2 + \[Nu]])) 2^(-(13/6) + Abs[1/2 + \[Nu]]) Sqrt[Pi] z^(-(1/2) - Abs[1/2 + \[Nu]]) Gamma[-(1/3)] Sign[1/2 + \[Nu]])/(3^(5/6) Gamma[1 - Abs[1/2 + \[Nu]]])) (2^(2/3) (AiryBi[(-(3/2)^(2/3)) z^(2/3)] - Sqrt[3] AiryAi[(-(3/2)^(2/3)) z^(2/3)] Sign[1/2 + \[Nu]]) Sum[((z^2)^k (-(1/3) - k + Abs[1/2 + \[Nu]])!)/ (4^k (k! (-(1/3) - 2 k + Abs[1/2 + \[Nu]])! Pochhammer[1/3, k] Pochhammer[1 - Abs[1/2 + \[Nu]], k])), {k, 0, -(1/3) + Abs[1/2 + \[Nu]]}] + 3^(1/6) z^(2/3) (Sqrt[3] AiryBiPrime[(-(3/2)^(2/3)) z^(2/3)] - 3 AiryAiPrime[(-(3/2)^(2/3)) z^(2/3)] Sign[1/2 + \[Nu]]) Sum[((z^2)^k (-(4/3) - k + Abs[1/2 + \[Nu]])!)/ (4^k (k! (-(4/3) - 2 k + Abs[1/2 + \[Nu]])! Pochhammer[4/3, k] Pochhammer[1 - Abs[1/2 + \[Nu]], k])), {k, 0, -(4/3) + Abs[1/2 + \[Nu]]}]) /; Element[Abs[1/2 + \[Nu]] - 1/3, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["13", "6"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]], " ", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], RowBox[List[SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["AiryBi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], "-", RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], " ", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["AiryAiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], " ", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], "-", FractionBox["1", "3"]]], "\[Element]", "Integers"]]]]]]

 MathML Form

 j ν ( z ) ( "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 1 3 ) ( sgn ( ν + 1 2 ) + 1 ) 2 "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 13 6 π z - "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 1 2 Γ ( - 1 3 ) sgn ( ν + 1 2 ) 3 5 / 6 Γ ( 1 - "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" ) ( 3 6 z 2 / 3 ( 3 Bi ( - ( 3 2 ) 2 / 3 z 2 / 3 ) - 3 Ai ( - ( 3 2 ) 2 / 3 z 2 / 3 ) sgn ( ν + 1 2 ) ) ( k = 0 "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 4 3 4 - k ( z 2 ) k ( - k + "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 4 3 ) ! k ! ( - 2 k + "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 4 3 ) ! ( 4 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], "k"], Pochhammer] ( 1 - ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ) + 2 2 / 3 ( Bi ( - ( 3 2 ) 2 / 3 z 2 / 3 ) - 3 Ai ( - ( 3 2 ) 2 / 3 z 2 / 3 ) sgn ( ν + 1 2 ) ) k = 0 "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 1 3 4 - k ( z 2 ) k ( - k + "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 1 3 ) ! k ! ( - 2 k + "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 1 3 ) ! ( 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "k"], Pochhammer] ( 1 - ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ) /; "\[LeftBracketingBar]" ν + 1 2 "\[RightBracketingBar]" - 1 3 TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition SphericalBesselJ ν z ν 1 2 -1 1 3 Sign ν 1 2 1 2 ν 1 2 -1 13 6 1 2 z -1 ν 1 2 -1 1 2 Gamma -1 1 3 Sign ν 1 2 3 5 6 Gamma 1 -1 ν 1 2 -1 3 1 6 z 2 3 3 1 2 AiryBiPrime -1 3 2 2 3 z 2 3 -1 3 AiryAiPrime -1 3 2 2 3 z 2 3 Sign ν 1 2 k 0 ν 1 2 -1 4 3 4 -1 k z 2 k -1 k ν 1 2 -1 4 3 k -2 k ν 1 2 -1 4 3 Pochhammer 4 3 k Pochhammer 1 -1 ν 1 2 k -1 2 2 3 AiryBi -1 3 2 2 3 z 2 3 -1 3 1 2 AiryAi -1 3 2 2 3 z 2 3 Sign ν 1 2 k 0 ν 1 2 -1 1 3 4 -1 k z 2 k -1 k ν 1 2 -1 1 3 k -2 k ν 1 2 -1 1 3 Pochhammer 1 3 k Pochhammer 1 -1 ν 1 2 k -1 ν 1 2 -1 1 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["13", "6"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]], " ", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["AiryBi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], "-", RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], " ", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["AiryAiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], " ", RowBox[List["Sign", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], "-", FractionBox["1", "3"]]], "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02