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 BesselK

 http://functions.wolfram.com/03.04.21.0047.01

 Input Form

 Integrate[BesselK[\[Nu], a z^r]^2, z] == -((4^(-1 - \[Nu]) Pi^2 z Csc[Pi \[Nu]]^2 (2^(1 + 2 \[Nu]) (a z^r)^(2 \[Nu]) (-1 + 4 r^2 \[Nu]^2) Gamma[1 - \[Nu]] Gamma[1 + \[Nu]] HypergeometricPFQ[{1/2, 1/(2 r)}, {1 + 1/(2 r), 1 - \[Nu], 1 + \[Nu]}, a^2 z^(2 r)] + (16^\[Nu] + 2^(1 + 4 \[Nu]) r \[Nu]) Gamma[1 + \[Nu]]^2 HypergeometricPFQ[{1/2 - \[Nu], 1/(2 r) - \[Nu]}, {1 - 2 \[Nu], 1 - \[Nu], 1 + 1/(2 r) - \[Nu]}, a^2 z^(2 r)] - (a z^r)^(4 \[Nu]) (-1 + 2 r \[Nu]) Gamma[1 - \[Nu]]^2 HypergeometricPFQ[{1/2 + \[Nu], 1/(2 r) + \[Nu]}, {1 + \[Nu], 1 + 1/(2 r) + \[Nu], 1 + 2 \[Nu]}, a^2 z^(2 r)]))/ (a z^r)^(2 \[Nu]))/((-1 + 4 r^2 \[Nu]^2) Gamma[1 - \[Nu]]^2 Gamma[1 + \[Nu]]^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], "2"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["r", "2"], " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["1", 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["(", RowBox[List[SuperscriptBox["16", "\[Nu]"], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]]], " ", "r", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["4", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["1", 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["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["r", "2"], " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]], ")"]]]]]]]]

 MathML Form

 K ν ( a z r ) 2 z - ( 4 - ν - 1 π 2 z ( a z r ) - 2 ν csc 2 ( π ν ) ( 2 2 ν + 1 ( 4 r 2 ν 2 - 1 ) Γ ( 1 - ν ) Γ ( ν + 1 ) 2 F 3 ( 1 2 , 1 2 r ; 1 + 1 2 r , 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["1", RowBox[List["2", " ", "r"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], 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] ( a z r ) 2 ν - ( 2 r ν - 1 ) Γ ( 1 - ν ) 2 2 F 3 ( ν + 1 2 , ν + 1 2 r ; ν + 1 , ν + 1 2 r + 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["\[Nu]", "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], 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["\[Nu]", "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "+", "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] ( a z r ) 4 ν + ( 2 4 ν + 1 r ν + 16 ν ) Γ ( ν + 1 ) 2 2 F 3 ( 1 2 - ν , 1 2 r - ν ; 1 - 2 ν , 1 - ν , - ν + 1 2 r + 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[FractionBox["1", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", 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["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "+", "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 r 2 ν 2 - 1 ) Γ ( 1 - ν ) 2 Γ ( ν + 1 ) 2 ) z BesselK ν a z r 2 -1 4 -1 ν -1 2 z a z r -2 ν ν 2 2 2 ν 1 4 r 2 ν 2 -1 Gamma 1 -1 ν Gamma ν 1 HypergeometricPFQ 1 2 1 2 r -1 1 1 2 r -1 1 -1 ν ν 1 a 2 z 2 r a z r 2 ν -1 2 r ν -1 Gamma 1 -1 ν 2 HypergeometricPFQ ν 1 2 ν 1 2 r -1 ν 1 ν 1 2 r -1 1 2 ν 1 a 2 z 2 r a z r 4 ν 2 4 ν 1 r ν 16 ν Gamma ν 1 2 HypergeometricPFQ 1 2 -1 ν 1 2 r -1 -1 ν 1 -1 2 ν 1 -1 ν -1 ν 1 2 r -1 1 a 2 z 2 r 4 r 2 ν 2 -1 Gamma 1 -1 ν 2 Gamma ν 1 2 -1 [/itex]

 Rule Form

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

 2001-10-29