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 BesselK

 http://functions.wolfram.com/03.04.21.0101.01

 Input Form

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

 Standard Form

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

 MathML Form

 z α - 1 I ν ( a z r ) K ν ( a z r ) z π z α csc ( π ν ) 2 Γ ( ν + 1 ) 2 ( Γ ( ν + 1 ) α Γ ( 1 - ν ) 2 F 3 ( 1 2 , α 2 r ; α 2 r + 1 , 1 - ν , ν + 1 ; a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], 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", 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 ν α + 2 r ν 2 F 3 ( ν + 1 2 , α 2 r + ν ; ν + 1 , α 2 r + ν + 1 , 2 ν + 1 ; a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[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 ν a z r BesselK ν a z r z α ν 2 Gamma ν 1 2 -1 Gamma ν 1 α Gamma 1 -1 ν -1 HypergeometricPFQ 1 2 α 2 r -1 α 2 r -1 1 1 -1 ν ν 1 a 2 z 2 r -1 4 -1 ν a z r 2 ν α 2 r ν -1 HypergeometricPFQ ν 1 2 α 2 r -1 ν ν 1 α 2 r -1 ν 1 2 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["\[Nu]_", ",", 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["\[Pi]", " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "\[Nu]"]]], " ", 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"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List["\[Alpha]", "+", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]]]]]]]

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

 2001-10-29