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 BesselI

 http://functions.wolfram.com/03.02.03.0011.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29

© 1998-2013 Wolfram Research, Inc.