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http://functions.wolfram.com/03.04.21.0098.01
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Integrate[(1/z) BesselI[\[Mu], a z] BesselK[\[Nu], a z], z] ==
2^(-1 - \[Mu] - \[Nu]) Pi (a z)^(\[Mu] - \[Nu]) Csc[Pi \[Nu]]
(4^\[Nu] Gamma[\[Mu] - \[Nu]] HypergeometricPFQRegularized[
{(\[Mu] - \[Nu])/2, (1/2) (1 + \[Mu] - \[Nu])},
{1 + \[Mu], 1 - \[Nu], 1 + \[Mu] - \[Nu]}, a^2 z^2] -
(a z)^(2 \[Nu]) Gamma[\[Mu] + \[Nu]] HypergeometricPFQRegularized[
{(\[Mu] + \[Nu])/2, (1/2) (1 + \[Mu] + \[Nu])},
{1 + \[Mu], 1 + \[Nu], 1 + \[Mu] + \[Nu]}, a^2 z^2])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", "z"], RowBox[List["BesselI", "[", RowBox[List["\[Mu]", ",", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]", "-", "\[Nu]"]]], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]]]]
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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> <mfrac> <mrow> <mrow> <msub> <mi> I </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 4 </mn> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mi> μ </mi> <mo> - </mo> <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[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", "1"]], ")"]]]], 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[RowBox[List["1", "-", "\[Nu]"]], 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[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> μ </mi> <mo> + </mo> <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[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], 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[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[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <ci> BesselI </ci> <ci> μ </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> BesselK </ci> <ci> ν </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <pi /> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <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='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <times /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> μ </ci> <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> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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