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 BesselK

 http://functions.wolfram.com/03.04.21.0068.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselK", "[", 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[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Mu]", "-", "\[Nu]"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", RowBox[List["\[Mu]", 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 MathML Form

 z α - 1 K μ ( a z r ) K ν ( a z r ) z 2 - μ - ν - 2 π 2 z α ( a z r ) - μ - ν csc ( π μ ) csc ( π ν ) ( 1 Γ ( ν + 1 ) ( ( a z r ) 2 ν ( 1 ( α + r ( μ + ν ) ) Γ ( μ + 1 ) ( ( a z r ) 2 μ 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] ) - 1 ( α - r μ + r ν ) Γ ( 1 - μ ) ( 4 μ 3 F 4 ( - μ 2 + ν 2 + 1 2 , - μ 2 + ν 2 + 1 , α 2 r + ν 2 - μ 2 ; 1 - μ , α 2 r + ν 2 - μ 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[RowBox[List["-", FractionBox["\[Mu]", "2"]]], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Mu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "-", FractionBox["\[Mu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[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] ) ) ) + 4 μ + ν ( α - r ( μ + ν ) ) Γ ( 1 - μ ) Γ ( 1 - ν ) 3 F 4 ( - μ 2 - ν 2 + 1 2 , - μ 2 - ν 2 + 1 , α 2 r - μ 2 - ν 2 ; 1 - μ , 1 - ν , - μ - ν + 1 , α 2 r - μ 2 - ν 2 + 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[RowBox[List["-", FractionBox["\[Mu]", "2"]]], "-", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[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["1", "-", "\[Mu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[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]], HypergeometricPFQ] - 4 ν ( a z r ) 2 μ ( α + r μ - r ν ) Γ ( μ + 1 ) 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[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]], HypergeometricPFQ] ) z z α -1 BesselK μ a z r BesselK ν a z r 2 -1 μ -1 ν -2 2 z α a z r -1 μ -1 ν μ ν 1 Gamma ν 1 -1 a z r 2 ν 1 α r μ ν Gamma μ 1 -1 a z r 2 μ 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 -1 1 α -1 r μ r ν Gamma 1 -1 μ -1 4 μ HypergeometricPFQ -1 μ 2 -1 ν 2 -1 1 2 -1 μ 2 -1 ν 2 -1 1 α 2 r -1 ν 2 -1 -1 μ 2 -1 1 -1 μ α 2 r -1 ν 2 -1 -1 μ 2 -1 1 ν 1 -1 μ ν 1 a 2 z 2 r 4 μ ν α -1 r μ ν Gamma 1 -1 μ Gamma 1 -1 ν -1 HypergeometricPFQ -1 μ 2 -1 -1 ν 2 -1 1 2 -1 μ 2 -1 -1 ν 2 -1 1 α 2 r -1 -1 μ 2 -1 -1 ν 2 -1 1 -1 μ 1 -1 ν -1 μ -1 ν 1 α 2 r -1 -1 μ 2 -1 -1 ν 2 -1 1 a 2 z 2 r -1 4 ν a z r 2 μ α r μ -1 r ν Gamma μ 1 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 [/itex]

 Rule Form

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

 2001-10-29