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 BesselK

 http://functions.wolfram.com/03.04.06.0043.01

 Input Form

 BesselK[n, z] == Subscript[F, Infinity][z, n] /; Subscript[F, m][z, n] == (-1)^(n - 1) Log[z/2] BesselI[n, z] + (1/2) Sum[(((-1)^k (n - k - 1)!)/k!) (z/2)^(2 k - n), {k, 0, n - 1}] + ((-1)^n/2) (z/2)^n Sum[((PolyGamma[k + 1] + PolyGamma[k + n + 1])/ (k! (k + n)!)) (z/2)^(2 k), {k, 0, m}] == (-1)^(n - 1) Log[z/2] BesselI[n, z] + (z^n BesselK[n, Sqrt[z^2]])/ (z^2)^(n/2) + (1/2) (-1)^n BesselI[n, z] Log[z^2/4] - (((-1)^n 2^(-3 - 2 m - n) z^(2 (1 + m) + n))/((1 + m)! (1 + m + n)!)) Log[z^2/4] HypergeometricPFQ[{1}, {2 + m, 2 + m + n}, z^2/4] - ((-1)^n/2) (z/2)^n MeijerG[{{1 + m, 1 + m}, {}}, {{1 + m, 1 + m}, {0, -n}}, z^2/4] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "n"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], " "]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2"], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "n"]], "/", "2"]]], " ", RowBox[List["BesselK", "[", RowBox[List["n", ",", SqrtBox[SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]], " ", RowBox[List["Log", "[", FractionBox[SuperscriptBox["z", "2"], "4"], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "m"]], "-", "n"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]]]], "+", "n"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m", "+", "n"]], ")"]], "!"]]]]], RowBox[List["Log", "[", FractionBox[SuperscriptBox["z", "2"], "4"], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "n"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2"], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "m"]], ",", RowBox[List["1", "+", "m"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "m"]], ",", RowBox[List["1", "+", "m"]]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "n"]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 K n ( z ) F ( z , n ) /; ( ( F m ( z , n ) ( - 1 ) n 2 ( z 2 ) n k = 0 m ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) k ! ( k + n ) ! ( z 2 ) 2 k + ( - 1 ) n - 1 log ( z 2 ) I n ( z ) + 1 2 k = 0 n - 1 ( - 1 ) k ( - k + n - 1 ) ! k ! ( z 2 ) 2 k - n z n ( z 2 ) - n 2 K n ( z 2 ) - log ( z 2 4 ) ( - 1 ) n 2 - 2 m - n - 3 z 2 ( m + 1 ) + n ( m + 1 ) ! ( m + n + 1 ) ! 1 F 2 ( 1 ; m + 2 , m + n + 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + ( - 1 ) n - 1 I n ( z ) log ( z 2 ) + 1 2 ( - 1 ) n log ( z 2 4 ) I n ( z ) - ( - 1 ) n 2 ( z 2 ) n G 2 , 4 2 , 2 ( z 2 4 m + 1 , m + 1 m + 1 , m + 1 , 0 , - n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["m", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["m", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition BesselK n z Subscript F z n Subscript F m z n -1 n 2 -1 z 2 -1 n k 0 m PolyGamma k 1 PolyGamma k n 1 k k n -1 z 2 -1 2 k -1 n -1 z 2 -1 BesselI n z 1 2 k 0 n -1 -1 k -1 k n -1 k -1 z 2 -1 2 k -1 n z n z 2 -1 n 2 -1 BesselK n z 2 1 2 -1 z 2 4 -1 -1 n 2 -2 m -1 n -3 z 2 m 1 n m 1 m n 1 -1 HypergeometricPFQ 1 m 2 m n 2 z 2 4 -1 -1 n -1 BesselI n z z 2 -1 1 2 -1 n z 2 4 -1 BesselI n z -1 -1 n 2 -1 z 2 -1 n MeijerG m 1 m 1 m 1 m 1 0 -1 n z 2 4 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "n"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]], RowBox[List["k", "!"]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["-", FractionBox["n", "2"]]]], " ", RowBox[List["BesselK", "[", RowBox[List["n", ",", SqrtBox[SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]], " ", RowBox[List["Log", "[", FractionBox[SuperscriptBox["z", "2"], "4"], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "m"]], "-", "n"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]]]], "+", "n"]]]]], ")"]], " ", RowBox[List["Log", "[", FractionBox[SuperscriptBox["z", "2"], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "n"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m", "+", "n"]], ")"]], "!"]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "m"]], ",", RowBox[List["1", "+", "m"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "m"]], ",", RowBox[List["1", "+", "m"]]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "n"]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02