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 BesselK

 http://functions.wolfram.com/03.04.21.0035.01

 Input Form

 Integrate[Cosh[a z^r] BesselK[\[Nu], a z^r], z] == (-(1/(-1 + r^2 \[Nu]^2))) ((2^(-1 - \[Nu]) z (4^\[Nu] (Gamma[\[Nu]] + r Gamma[1 + \[Nu]]) HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/(2 r) - \[Nu]/2}, {1/2, 1/2 - \[Nu], 1 - \[Nu], 1 + 1/(2 r) - \[Nu]/2}, a^2 z^(2 r)] + (a z^r)^(2 \[Nu]) (r Gamma[1 - \[Nu]] + Gamma[-\[Nu]]) HypergeometricPFQ[{1/4 + \[Nu]/2, 3/4 + \[Nu]/2, 1/(2 r) + \[Nu]/2}, {1/2, 1 + 1/(2 r) + \[Nu]/2, 1/2 + \[Nu], 1 + \[Nu]}, a^2 z^(2 r)]))/ (a z^r)^\[Nu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cosh", "[", 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[RowBox[List["-", FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["r", "2"], " ", SuperscriptBox["\[Nu]", "2"]]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "\[Nu]", "]"]], "+", RowBox[List["r", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["r", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]], "+", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", 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["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "\[Nu]", "]"]], "+", RowBox[List["r", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["r", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]], "+", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["r", "2"], " ", SuperscriptBox["\[Nu]", "2"]]]]]]]]]]]]

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

 2001-10-29