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; }

 BesselJ

 http://functions.wolfram.com/03.01.21.0083.01

 Input Form

 Integrate[t^(\[Alpha] - 1) BesselJ[\[Lambda], a t] BesselJ[\[Mu], b t] BesselJ[\[Nu], c t], {t, 0, Infinity}] == 2^(\[Alpha] - 1) a^\[Lambda] b^\[Mu] c^(-\[Alpha] - \[Lambda] - \[Mu]) (Gamma[(\[Alpha] + \[Lambda] + \[Mu] + \[Nu])/2]/ (Gamma[\[Lambda] + 1] Gamma[\[Mu] + 1] Gamma[1 - (\[Alpha] + \[Lambda] + \[Mu] - \[Nu])/2])) HypergeometricPFQ[{{(\[Alpha] + \[Lambda] + \[Mu] + \[Nu])/2, (\[Alpha] + \[Lambda] + \[Mu] - \[Nu])/2}, {}, {}}, {{}, {\[Lambda] + 1}, {\[Mu] + 1}}, a^2/c^2, b^2/c^2] /; Element[a, Reals] && Element[b, Reals] && Element[c, Reals] && a > 0 && b > 0 && c > a + b && Re[\[Alpha] + \[Lambda] + \[Mu] + \[Nu]] > 0 && Re[\[Alpha]] < 5/2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["BesselJ", "[", RowBox[List["\[Lambda]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["b", " ", "t"]]]], "]"]], RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["a", "\[Lambda]"], SuperscriptBox["b", "\[Mu]"], SuperscriptBox["c", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Lambda]", "-", "\[Mu]"]]], FractionBox[RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "2"], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox[RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], "2"]]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["\[Lambda]", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", "1"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox[StyleBox["a", "TI"], "2"], SuperscriptBox["c", "2"]], ",", FractionBox[SuperscriptBox[StyleBox["b", "TI"], "2"], SuperscriptBox["c", "2"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "\[And]", RowBox[List["b", "\[Element]", "Reals"]], "\[And]", RowBox[List["c", "\[Element]", "Reals"]], "\[And]", RowBox[List["a", ">", "0"]], "\[And]", RowBox[List["b", ">", "0"]], "\[And]", RowBox[List["c", ">", RowBox[List["a", "+", "b"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["5", "2"]]]]]]]]]

 MathML Form

 0 t α - 1 J λ ( a t ) J μ ( b t ) J ν ( c t ) t 2 α - 1 a λ b μ c - α - λ - μ Γ ( 1 2 ( α + λ + μ + ν ) ) Γ ( λ + 1 ) Γ ( μ + 1 ) Γ ( 1 - 1 2 ( α + λ + μ - ν ) ) F 0 1 1 2 0 0 ( α + λ + μ + ν 2 , α + λ + μ - ν 2 ; ; ; ; λ + 1 ; μ + 1 ; a 2 c 2 , b 2 c 2 ) /; a TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] b TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] c TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] a > 0 b > 0 c > a + b Re ( α + λ + μ + ν ) > 0 Re ( α ) < 5 2 FormBox RowBox RowBox RowBox SubsuperscriptBox 0 RowBox RowBox SuperscriptBox t RowBox α - 1 RowBox SubscriptBox J λ ( RowBox a t ) RowBox SubscriptBox J μ ( RowBox b t ) RowBox SubscriptBox J ν ( RowBox c t ) RowBox t RowBox FractionBox RowBox SuperscriptBox 2 RowBox α - 1 SuperscriptBox a λ SuperscriptBox b μ SuperscriptBox c RowBox RowBox - α - λ - μ RowBox Γ ( RowBox FractionBox 1 2 RowBox ( RowBox α + λ + μ + ν ) ) RowBox RowBox Γ ( RowBox λ + 1 ) RowBox Γ ( RowBox μ + 1 ) RowBox Γ ( RowBox 1 - RowBox FractionBox 1 2 RowBox ( RowBox α + λ + μ - ν ) ) RowBox SubsuperscriptBox F RowBox 0 1 1 RowBox 2 0 0 [ RowBox GridBox ErrorBox RowBox FractionBox RowBox α + λ + μ + ν 2 , RowBox RowBox RowBox FractionBox RowBox α + λ + μ - ν 2 ; ; ; RowBox ; RowBox λ + 1 ; RowBox μ + 1 ; FractionBox SuperscriptBox a 2 SuperscriptBox c 2 , FractionBox SuperscriptBox b 2 SuperscriptBox c 2 ] /; RowBox RowBox a TagBox Function RowBox b TagBox Function RowBox c TagBox Function RowBox a > 0 RowBox b > 0 RowBox c > RowBox a + b RowBox RowBox Re ( RowBox α + λ + μ + ν ) > 0 RowBox RowBox Re ( α ) < FractionBox 5 2 TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Lambda]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["c_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["a", "\[Lambda]"], " ", SuperscriptBox["b", "\[Mu]"], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Lambda]", "-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["\[Lambda]", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", "1"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["c", "2"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["b", "\[Element]", "Reals"]], "&&", RowBox[List["c", "\[Element]", "Reals"]], "&&", RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List["c", ">", RowBox[List["a", "+", "b"]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["5", "2"]]]]]]]]]]]

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

 2007-05-02