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 BesselI

 http://functions.wolfram.com/03.02.03.0012.01

 Input Form

 BesselI[\[Nu], z] == ((Sign[\[Nu]] 2^(Abs[\[Nu]] - 7/3) Gamma[-(2/3)])/ (z^Abs[\[Nu]] (3 3^(5/6) Gamma[1 - Abs[\[Nu]]]))) (9 z^(4/3) (Sqrt[3] AiryAi[(3/2)^(2/3) z^(2/3)] + Sign[\[Nu]] AiryBi[(3/2)^(2/3) z^(2/3)]) Sum[((Abs[\[Nu]] - k - 5/3)!/(k! (Abs[\[Nu]] - 2 k - 5/3)! Pochhammer[5/3, k] Pochhammer[1 - Abs[\[Nu]], k])) (-(z^2/4))^k, {k, 0, Abs[\[Nu]] - 5/3}] - 4 2^(1/3) 3^(1/6) (3 AiryAiPrime[(3/2)^(2/3) z^(2/3)] + Sign[\[Nu]] Sqrt[3] AiryBiPrime[(3/2)^(2/3) z^(2/3)]) Sum[((Abs[\[Nu]] - k - 2/3)!/(k! (Abs[\[Nu]] - 2 k - 2/3)! Pochhammer[2/3, k] Pochhammer[1 - Abs[\[Nu]], k])) (-(z^2/4))^k, {k, 0, Abs[\[Nu]] - 2/3}]) /; Element[Abs[\[Nu]] - 2/3, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["7", "3"]]]], SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["2", "3"]]], "]"]], " "]], RowBox[List["3", " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["9", " ", SuperscriptBox["z", RowBox[List["4", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[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[RowBox[List["Sign", "[", "\[Nu]", "]"]], 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["5", "3"]]]], RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["5", "3"]]], ")"]], "!"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["5", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "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["4", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], 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["2", "3"]]]], " ", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["2", "3"]]], ")"]], "!"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["2", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "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["2", "3"]]], ",", "Integers"]], "]"]]]]]]

 MathML Form

 I ν ( z ) sgn ( ν ) 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 7 3 z - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" Γ ( - 2 3 ) 3 3 5 / 6 Γ ( 1 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ( 9 z 4 / 3 ( 3 Ai ( ( 3 2 ) 2 / 3 z 2 / 3 ) + sgn ( ν ) Bi ( ( 3 2 ) 2 / 3 z 2 / 3 ) ) k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 5 3 ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 5 3 ) ! k ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 5 3 ) ! ( 5 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( - z 2 4 ) k - 4 2 3 3 6 ( 3 Ai ( ( 3 2 ) 2 / 3 z 2 / 3 ) + 3 sgn ( ν ) Bi ( ( 3 2 ) 2 / 3 z 2 / 3 ) ) k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 2 3 ) ! k ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 2 3 ) ! ( 2 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["2", "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]" - 2 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition BesselI ν z Sign ν 2 ν -1 7 3 z -1 ν Gamma -1 2 3 3 3 5 6 Gamma 1 -1 ν -1 9 z 4 3 3 1 2 AiryAi 3 2 2 3 z 2 3 Sign ν AiryBi 3 2 2 3 z 2 3 k 0 ν -1 5 3 ν -1 k -1 5 3 k ν -1 2 k -1 5 3 Pochhammer 5 3 k Pochhammer 1 -1 ν k -1 -1 z 2 4 -1 k -1 4 2 1 3 3 1 6 3 AiryAiPrime 3 2 2 3 z 2 3 3 1 2 Sign ν AiryBiPrime 3 2 2 3 z 2 3 k 0 ν -1 2 3 ν -1 k -1 2 3 k ν -1 2 k -1 2 3 Pochhammer 2 3 k Pochhammer 1 -1 ν k -1 -1 z 2 4 -1 k ν -1 2 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[RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["7", "3"]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["2", "3"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["9", " ", SuperscriptBox["z", RowBox[List["4", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[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[RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", 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["5", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["5", "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["5", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]]]]]], "-", RowBox[List["4", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", 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["2", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["2", "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["2", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List["3", " ", 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["2", "3"]]], "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29