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/03.01.21.0070.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselJ[\[Mu], a z^r] BesselJ[\[Nu], a z^r], z] == (1/r) (2^(-1 - \[Mu] - \[Nu]) z^\[Alpha] (a z^r)^(\[Mu] + \[Nu]) Gamma[1 + \[Mu] + \[Nu]] Gamma[(\[Alpha] + r (\[Mu] + \[Nu]))/(2 r)] HypergeometricPFQRegularized[{(1/2) (1 + \[Mu] + \[Nu]), (1/2) (2 + \[Mu] + \[Nu]), (\[Alpha] + r (\[Mu] + \[Nu]))/(2 r)}, {1 + \[Mu], (\[Alpha] + r (2 + \[Mu] + \[Nu]))/(2 r), 1 + \[Nu], 1 + \[Mu] + \[Nu]}, (-a^2) z^(2 r)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "r"], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]", "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["\[Mu]", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 J μ ( a z r ) J ν ( a z r ) z 1 r ( 2 - μ - ν - 1 z α ( a z r ) μ + ν Γ ( μ + ν + 1 ) Γ ( α + r ( μ + ν ) 2 r ) 3 F ~ 4 ( 1 2 ( μ + ν + 1 ) , 1 2 ( μ + ν + 2 ) , α + r ( μ + ν ) 2 r ; μ + 1 , α + r ( μ + ν + 2 ) 2 r , ν + 1 , μ + ν + 1 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Mu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) z z α -1 BesselJ μ a z r BesselJ ν a z r 1 r -1 2 -1 μ -1 ν -1 z α a z r μ ν Gamma μ ν 1 Gamma α r μ ν 2 r -1 HypergeometricPFQRegularized 1 2 μ ν 1 1 2 μ ν 2 α r μ ν 2 r -1 μ 1 α r μ ν 2 2 r -1 ν 1 μ ν 1 -1 a 2 z 2 r [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]", "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["\[Mu]", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "r"]]]]]

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

 2001-10-29