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; }

 BesselI

 http://functions.wolfram.com/03.02.26.0083.01

 Input Form

 BesselI[-1 - n - \[Nu], Sqrt[z]] BesselI[\[Nu], Sqrt[z]] == (-(-1)^Floor[n/2]) Sqrt[2 Pi] MeijerG[{{0, 1/2}, {1/4}}, {{(1 + n)/2}, {\[Nu] + (1 + n)/2, -((1 + n)/2), -((1 + n)/2) - \[Nu], 1/4}}, z] + (2/Sqrt[Pi]) Sum[(((-1)^Floor[(1 + n)/2] z^((-1 + 2 k - n)/2) Gamma[1/2 + k - n + Floor[n/2]])/(k! Gamma[k - n - \[Nu]] Gamma[1 + k + \[Nu]])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "n", "-", "\[Nu]"]], ",", SqrtBox["z"]]], "]"]], RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", SqrtBox["z"]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], " ", SqrtBox[RowBox[List["2", "\[Pi]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "+", "n"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], ",", RowBox[List["-", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], "-", "\[Nu]"]], ",", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["2", SqrtBox["\[Pi]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["z", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "k"]], "-", "n"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 I - n - ν - 1 ( z ) I ν ( z ) 2 π k = 0 n 2 ( - 1 ) n + 1 2 z 1 2 ( 2 k - n - 1 ) Γ ( k - n + n 2 + 1 2 ) ( 1 - k + n 2 ) n - n 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "+", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]], ")"]], RowBox[List["n", "-", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]]], Pochhammer] k ! Γ ( k - n - ν ) Γ ( k + ν + 1 ) - ( - 1 ) n 2 2 π G 3 , 5 1 , 2 ( z 0 , 1 2 , 1 4 n + 1 2 , n + 1 2 + ν , - 1 2 ( n + 1 ) , - 1 2 ( n + 1 ) - ν , 1 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["n", "+", "1"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], "+", "\[Nu]"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "-", "\[Nu]"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Condition BesselI -1 n -1 ν -1 z 1 2 BesselI ν z 1 2 2 1 2 -1 k 0 n 2 -1 -1 n 1 2 -1 z 1 2 2 k -1 n -1 Gamma k -1 n n 2 -1 1 2 Pochhammer 1 -1 k n 2 -1 n -1 n 2 -1 k Gamma k -1 n -1 ν Gamma k ν 1 -1 -1 -1 n 2 -1 2 1 2 MeijerG 0 1 2 1 4 n 1 2 -1 n 1 2 -1 ν -1 1 2 n 1 -1 1 2 n 1 -1 ν 1 4 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "n_", "-", "\[Nu]_"]], ",", SqrtBox["z_"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", SqrtBox["z_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "+", "n"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], "-", "\[Nu]"]], ",", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "k"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]]]]]]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02