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 BesselK

 http://functions.wolfram.com/03.04.21.0027.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a_"]], " ", SuperscriptBox["z_", "r_"]]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", "r"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "r"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], RowBox[List["\[Alpha]", "-", RowBox[List["r", " ", "\[Nu]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", "r"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", "r"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]]]]], ")"]]]]]]]]

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

 2001-10-29