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; }

 Fibonacci

 http://functions.wolfram.com/07.06.06.0014.01

 Input Form

 Fibonacci[\[Nu], z] == ((\[Nu] (2 I Cos[Pi \[Nu]] + Sin[Pi \[Nu]]))/2) E^((I Pi \[Nu])/2) Sum[((Pochhammer[1 - \[Nu], k] Pochhammer[1 + \[Nu], k])/ (k! Pochhammer[3/2, k])) (-(I/4))^k (z + 2 I)^k, {k, 0, Infinity}] - ((I Sin[Pi \[Nu]])/(2 Sqrt[(-I) (z + 2 I)])) E^((I Pi \[Nu])/2) Sum[((Pochhammer[1/2 - \[Nu], k] Pochhammer[1/2 + \[Nu], k])/ (k! Pochhammer[1/2, k])) (-(I/4))^k (z + 2 I)^k, {k, 0, Infinity}] /; Abs[z + 2 I] < 4

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Nu]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]]]], RowBox[List[" ", "2"]]], SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[ImaginaryI]", "4"]]], ")"]], "k"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", "\[ImaginaryI]"]]]], ")"]], "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", "\[ImaginaryI]"]]]], ")"]]]]]]]], SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[ImaginaryI]", "4"]]], ")"]], "k"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", "\[ImaginaryI]"]]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "+", RowBox[List["2", "\[ImaginaryI]"]]]], "]"]], "<", "4"]]]]]]

 MathML Form

 F TagBox["F", Fibonacci] ν ( z ) ν ( 2 cos ( π ν ) + sin ( π ν ) ) 2 π ν 2 k = 0 ( - 4 ) k ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( 2 + z ) k - sin ( π ν ) 2 - ( z + 2 ) π ν 2 k = 0 ( - 4 ) k ( 1 2 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] k ! ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 2 + z ) k /; "\[LeftBracketingBar]" z + 2 "\[RightBracketingBar]" < 4 Condition Fibonacci ν z ν 2 ν ν 2 -1 ν 2 -1 k 0 -1 4 -1 k Pochhammer 1 -1 ν k Pochhammer ν 1 k k Pochhammer 3 2 k -1 2 z k -1 ν 2 -1 z 2 1 2 -1 ν 2 -1 k 0 -1 4 -1 k Pochhammer 1 2 -1 ν k Pochhammer ν 1 2 k k Pochhammer 1 2 k -1 2 z k z 2 4 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[ImaginaryI]", "4"]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[ImaginaryI]", "4"]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], "]"]], "<", "4"]]]]]]]]

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

 2001-10-29