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 BesselI

 http://functions.wolfram.com/03.02.06.0034.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", 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["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "i"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "i"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["x", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["i", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "x"]], "]"]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "i"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselI", "[", 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["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "i"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "i"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "x", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["i", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "x"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "i"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]]]]]]]]]], 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