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http://functions.wolfram.com/03.09.21.0020.01
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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
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> μ </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mi> μ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> b </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> <apply> <ci> StruveH </ci> <ci> μ </ci> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> α </ci> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> α </ci> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <plus /> <ci> μ </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </list> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <lt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
