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 BesselK

 http://functions.wolfram.com/03.04.21.0100.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselI[\[Mu], a z^r] BesselK[\[Nu], a z^r], z] == (1/Gamma[1 + \[Mu]]) (2^(-1 - \[Mu] - \[Nu]) Pi z^\[Alpha] (a z^r)^(\[Mu] - \[Nu]) Csc[Pi \[Nu]] ((4^\[Nu] HypergeometricPFQ[{1/2 + \[Mu]/2 - \[Nu]/2, 1 + \[Mu]/2 - \[Nu]/2, \[Alpha]/(2 r) + \[Mu]/2 - \[Nu]/2}, {1 + \[Mu], 1 - \[Nu], 1 + \[Mu] - \[Nu], 1 + \[Alpha]/(2 r) + \[Mu]/2 - \[Nu]/2}, a^2 z^(2 r)])/((\[Alpha] + r (\[Mu] - \[Nu])) Gamma[1 - \[Nu]]) - ((a z^r)^(2 \[Nu]) HypergeometricPFQ[ {1/2 + \[Mu]/2 + \[Nu]/2, 1 + \[Mu]/2 + \[Nu]/2, \[Alpha]/(2 r) + \[Mu]/2 + \[Nu]/2}, {1 + \[Mu], 1 + \[Alpha]/(2 r) + \[Mu]/2 + \[Nu]/2, 1 + \[Nu], 1 + \[Mu] + \[Nu]}, a^2 z^(2 r)])/((\[Alpha] + r (\[Mu] + \[Nu])) Gamma[1 + \[Nu]])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Mu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]", "-", "\[Nu]"]]], " ", "\[Pi]", " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 I μ ( a z r ) K ν ( a z r ) z 1 Γ ( μ + 1 ) ( 2 - μ - ν - 1 π z α ( a z r ) μ - ν csc ( π ν ) ( 4 ν ( α + r ( μ - ν ) ) Γ ( 1 - ν ) 3 F 4 ( μ 2 - ν 2 + 1 2 , μ 2 - ν 2 + 1 , α 2 r + μ 2 - ν 2 ; μ + 1 , 1 - ν , μ - ν + 1 , α 2 r + μ 2 - ν 2 + 1 ; a 2 z 2 r ) 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["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Mu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "-", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]] - ( a z r ) 2 ν ( α + r ( μ + ν ) ) Γ ( ν + 1 ) 3 F 4 ( μ 2 + ν 2 + 1 2 , μ 2 + ν 2 + 1 , α 2 r + μ 2 + ν 2 ; μ + 1 , α 2 r + μ 2 + ν 2 + 1 , ν + 1 , μ + ν + 1 ; a 2 z 2 r ) TagBox[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["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Mu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) z z α -1 BesselI μ a z r BesselK ν a z r 1 Gamma μ 1 -1 2 -1 μ -1 ν -1 z α a z r μ -1 ν ν 4 ν α r μ -1 ν Gamma 1 -1 ν -1 HypergeometricPFQ μ 2 -1 -1 ν 2 -1 1 2 μ 2 -1 -1 ν 2 -1 1 α 2 r -1 μ 2 -1 -1 ν 2 -1 μ 1 1 -1 ν μ -1 ν 1 α 2 r -1 μ 2 -1 -1 ν 2 -1 1 a 2 z 2 r -1 a z r 2 ν α r μ ν Gamma ν 1 -1 HypergeometricPFQ μ 2 -1 ν 2 -1 1 2 μ 2 -1 ν 2 -1 1 α 2 r -1 μ 2 -1 ν 2 -1 μ 1 α 2 r -1 μ 2 -1 ν 2 -1 1 ν 1 μ ν 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["BesselI", "[", RowBox[List["\[Mu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]], " ", RowBox[List["BesselK", "[", 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]"]]], " ", "\[Pi]", " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Mu]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]]]]]]]

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

 2001-10-29