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 BesselI

 http://functions.wolfram.com/03.02.20.0020.01

 Input Form

 Derivative[1, 0][BesselI][-n - 1/2, z] == (((-1)^n 2 (2 z)^(-(1/2) - n))/(n! Sqrt[Pi])) Sum[2^(2 k) Binomial[n, 2 k] (2 n - 2 k)! ((PolyGamma[k + 1/2] - PolyGamma[k - n + 1/2]) Cosh[z] + Cosh[z] CoshIntegral[2 z] - Sinh[z] SinhIntegral[2 z]) z^(2 k), {k, 0, Floor[n/2]}] + (((-1)^n 2 (2 z)^(1/2 - n))/(n! Sqrt[Pi])) Sum[2^(2 k) Binomial[n, 2 k + 1] (2 n - 2 k - 1)! ((-(PolyGamma[k + 3/2] - PolyGamma[k - n + 1/2])) Sinh[z] + Cosh[z] SinhIntegral[2 z] - Sinh[z] CoshIntegral[2 z]) z^(2 k), {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "BesselI", "]"]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2", SuperscriptBox[RowBox[List["(", RowBox[List["2", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]]]], RowBox[List[RowBox[List["n", "!"]], SqrtBox["\[Pi]"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", "k"]]], RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", "k"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", "k"]]]], ")"]], "!"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["CoshIntegral", "[", RowBox[List["2", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], RowBox[List["SinhIntegral", "[", RowBox[List["2", "z"]], "]"]]]]]], ")"]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2", SuperscriptBox[RowBox[List["(", RowBox[List["2", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "n"]]]]], RowBox[List[RowBox[List["n", "!"]], SqrtBox["\[Pi]"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]]]], RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List["SinhIntegral", "[", RowBox[List["2", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], RowBox[List["CoshIntegral", "[", RowBox[List["2", "z"]], "]"]]]]]], ")"]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 I TagBox["I", BesselJ] - n - 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) ( - 1 ) n 2 ( 2 z ) 1 2 - n n ! π k = 0 n - 1 2 2 2 k ( n 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 2 k + 2 n - 1 ) ! ( - ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sinh ( z ) - Chi ( 2 z ) sinh ( z ) + cosh ( z ) Shi ( 2 z ) ) z 2 k + ( - 1 ) n 2 ( 2 z ) - n - 1 2 n ! π k = 0 n 2 2 2 k ( n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 n - 2 k ) ! ( cosh ( z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + cosh ( z ) Chi ( 2 z ) - sinh ( z ) Shi ( 2 z ) ) z 2 k /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript BesselJ -1 n -1 1 2 z -1 n 2 2 z 1 2 -1 n n 1 2 -1 k 0 n -1 2 -1 2 2 k Binomial n 2 k 1 -2 k 2 n -1 -1 PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 z -1 CoshIntegral 2 z z z SinhIntegral 2 z z 2 k -1 n 2 2 z -1 n -1 1 2 n 1 2 -1 k 0 n 2 -1 2 2 k Binomial n 2 k 2 n -1 2 k z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 z CoshIntegral 2 z -1 z SinhIntegral 2 z z 2 k n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["BesselI", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "-", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]]]], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]]]]], "/;", 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