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http://functions.wolfram.com/03.02.21.0040.01
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Integrate[z^(\[Alpha] - 1) Cosh[a z] BesselI[\[Nu], a z], z] ==
(z^\[Alpha] (a z)^\[Nu] HypergeometricPFQ[{1/4 + \[Nu]/2, 3/4 + \[Nu]/2,
\[Alpha]/2 + \[Nu]/2}, {1/2, 1 + \[Alpha]/2 + \[Nu]/2, 1/2 + \[Nu],
1 + \[Nu]}, a^2 z^2])/2^\[Nu]/((\[Alpha] + \[Nu]) Gamma[1 + \[Nu]])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 4 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> α </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> α </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> α </ci> <ci> ν </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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