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 StruveH

 http://functions.wolfram.com/03.09.21.0020.01

 Input Form

 Integrate[t^(\[Alpha] - 1) BesselJ[\[Nu], a t] StruveH[\[Mu], b t], {t, 0, Infinity}] == 2^\[Alpha] a^(-1 - \[Alpha] - \[Mu]) b^(1 + \[Mu]) Gamma[(1/2) (1 + \[Alpha] + \[Mu] + \[Nu])] (HypergeometricPFQ[{1, (1/2) (1 + \[Alpha] + \[Mu] + \[Nu]), (1/2) (1 + \[Alpha] + \[Mu] - \[Nu])}, {3/2 + \[Mu], 3/2}, b^2/a^2]/ (Sqrt[Pi] Gamma[3/2 + \[Mu]] Gamma[(1/2) (1 - \[Alpha] - \[Mu] + \[Nu])])) /; a > 0 && b > 0 && Re[\[Nu] + \[Mu]] > -1 && Re[\[Alpha]] < 1

 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["\[Nu]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", RowBox[List["b", " ", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]", "-", "\[Mu]"]]], " ", SuperscriptBox["b", RowBox[List["1", "+", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], ",", FractionBox["3", "2"]]], "}"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]], "]"]], "/", RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["a", ">", "0"]], "\[And]", RowBox[List["b", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "1"]]]]]]]]

 MathML Form

 0 t α - 1 J ν ( a t ) H StruveH μ ( b t ) t 2 α a - α - μ - 1 b μ + 1 π Γ ( μ + 3 2 ) Γ ( 1 2 ( - α - μ + ν + 1 ) ) Γ ( 1 2 ( α + μ + ν + 1 ) ) 3 F 2 ( 1 , 1 2 ( α + μ + ν + 1 ) , 1 2 ( α + μ - ν + 1 ) ; μ + 3 2 , 3 2 ; b 2 a 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "-", "\[Nu]", "+", "1"]], ")"]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Mu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; a > 0 b > 0 Re ( μ + ν ) > - 1 Re ( α ) < 1 Condition t 0 t α -1 BesselJ ν a t StruveH μ b t 2 α a -1 α -1 μ -1 b μ 1 1 2 Gamma μ 3 2 Gamma 1 2 -1 α -1 μ ν 1 -1 Gamma 1 2 α μ ν 1 HypergeometricPFQ 1 1 2 α μ ν 1 1 2 α μ -1 ν 1 μ 3 2 3 2 b 2 a 2 -1 a 0 b 0 μ ν -1 α 1 [/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["\[Nu]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Mu]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]", "-", "\[Mu]"]]], " ", SuperscriptBox["b", RowBox[List["1", "+", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], ",", FractionBox["3", "2"]]], "}"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "1"]]]]]]]]]]

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

 2001-10-29