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 BesselK

 http://functions.wolfram.com/03.04.20.0019.01

 Input Form

 Derivative[1, 0][BesselK][n + 1/2, z] == (((-1)^(n + 1) Pi)/2) (BesselI[n + 1/2, z] + BesselI[-n - 1/2, z]) (CoshIntegral[2 z] - SinhIntegral[2 z]) + ((n + 1)!/2) Sum[(1/(k! (n - k + 1))) (z/2)^(k - n - 1) BesselK[k - 1/2, z], {k, 0, n}] + (((-1)^n Sqrt[Pi z] (n + 1)!)/2) Sum[(1/((n - k + 1)! k)) (-(2/z))^k (BesselI[-k + n + 1/2, z] + BesselI[k - n - 1/2, z]) Sum[(z^p/p!) BesselK[p - 1/2, 2 z], {p, 0, k - 1}], {k, 1, n + 1}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "BesselK", "]"]], "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", "\[Pi]", " "]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k", "+", "1"]], ")"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["k", "-", "n", "-", "1"]]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List["\[Pi]", " ", "z"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "+", "1"]], ")"]], "!"]], " ", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", "z"]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["k", "-", "n", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox["z", "p"], RowBox[List["p", "!"]]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 K TagBox["K", BesselK] n + 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) ( - 1 ) n + 1 π 2 ( Chi ( 2 z ) - Shi ( 2 z ) ) ( I n + 1 2 ( z ) + I - n - 1 2 ( z ) ) + 1 2 ( n + 1 ) ! k = 0 n 1 k ! ( n - k + 1 ) ( z 2 ) k - n - 1 K k - 1 2 ( z ) + ( - 1 ) n π z ( n + 1 ) ! 2 k = 1 n + 1 1 ( n - k + 1 ) ! k ( - 2 z ) k ( I n - k + 1 2 ( z ) + I k - n - 1 2 ( z ) ) p = 0 k - 1 z p p ! K p - 1 2 ( 2 z ) /; n Condition 1 0 Subscript BesselK K n 1 2 z -1 n 1 2 -1 CoshIntegral 2 z -1 SinhIntegral 2 z BesselI n 1 2 z Subscript -1 n -1 1 2 z 1 2 n 1 k 0 n 1 k n -1 k 1 -1 z 2 -1 k -1 n -1 BesselK k -1 1 2 z -1 n z 1 2 n 1 2 -1 k 1 n 1 1 n -1 k 1 k -1 -1 2 z -1 k BesselI n -1 k 1 2 z BesselI k -1 n -1 1 2 z p 0 k -1 z p p -1 BesselK p -1 1 2 2 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["BesselK", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["n_", "+", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", "\[Pi]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["k", "-", "n", "-", "1"]]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k", "+", "1"]], ")"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List["\[Pi]", " ", "z"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "+", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", "z"]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["k", "-", "n", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox["z", "p"], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]], RowBox[List["p", "!"]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "+", "1"]], ")"]], "!"]], " ", "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02