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 Hypergeometric0F1Regularized

 http://functions.wolfram.com/07.18.20.0016.01

 Input Form

 Derivative[1, 0][Hypergeometric0F1Regularized][n + 1/2, z] == KroneckerDelta[n] ((Cosh[2 Sqrt[z]] CoshIntegral[4 Sqrt[z]] - Sinh[2 Sqrt[z]] SinhIntegral[4 Sqrt[z]])/Sqrt[Pi]) - (1/2) z^((1 - 2 n)/4) Log[z] BesselI[-(1/2) + n, 2 Sqrt[z]] - ((-1)^n/((n - 1)! Sqrt[Pi])) 2^(2 - 2 n) z^(1/2 - n) (Sum[Binomial[n - 1, 2 k] (2 n - 2 k - 2)! ((-PolyGamma[k + 1/2] + PolyGamma[k - n + 3/2]) Sinh[2 Sqrt[z]] + Sinh[2 Sqrt[z]] CoshIntegral[4 Sqrt[z]] - Cosh[2 Sqrt[z]] SinhIntegral[4 Sqrt[z]]) (16 z)^k, {k, 0, Floor[(n - 1)/2]}] + 4 z^(1/2) Sum[Binomial[n - 1, 2 k + 1] (2 n - 2 k - 3)! ((PolyGamma[k + 3/2] - PolyGamma[k - n + 3/2]) Cosh[2 Sqrt[z]] - Cosh[2 Sqrt[z]] CoshIntegral[4 Sqrt[z]] + Sinh[2 Sqrt[z]] SinhIntegral[4 Sqrt[z]]) (16 z)^k, {k, 0, Floor[n/2] - 1}]) /; Element[n, Integers] && n >= 0 && z != 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "Hypergeometric0F1Regularized", "]"]], "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], FractionBox[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]]]], SqrtBox["\[Pi]"]]]], "-", RowBox[List[FractionBox["1", "2"], SuperscriptBox["z", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["2", "n"]]]], "4"], " "]]], " ", RowBox[List["Log", "[", "z", "]"]], RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "-", " ", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], SqrtBox["\[Pi]"], " "]]], SuperscriptBox["2", RowBox[List["2", "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "n", " "]]], " ", RowBox[List["(", " ", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", "k"]], "-", "2"]], ")"]], "!"]], " ", RowBox[List["(", 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["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", SqrtBox["z"]]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["16", "z"]], ")"]], "k"]]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", FractionBox["1", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", "k"]], "-", "3"]], ")"]], "!"]], " ", 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["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", SqrtBox["z"]]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["16", "z"]], ")"]], "k"]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["z", "\[NotEqual]", "0"]]]]]]]]

 MathML Form

 0 F ~ 1 ( 1 , 0 ) ( TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] ; n + 1 2 ; z TagBox["z", Hypergeometric0F1, Rule[Editable, True]] ) δ KroneckerDelta n cosh ( 2 z ) Chi ( 4 z ) - sinh ( 2 z ) Shi ( 4 z ) π - 1 2 z 1 - 2 n 4 log ( z ) I n - 1 2 ( 2 z ) - ( - 1 ) n 2 2 - 2 n ( n - 1 ) ! π z 1 2 - n ( 4 z k = 0 n 2 - 1 ( n - 1 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - 2 k + 2 n - 3 ) ! ( - cosh ( 2 z ) Chi ( 4 z ) + cosh ( 2 z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 3 2 ) ) + sinh ( 2 z ) Shi ( 4 z ) ) ( 16 z ) k + k = 0 n - 1 2 ( n - 1 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - 2 k + 2 n - 2 ) ! ( Chi ( 4 z ) sinh ( 2 z ) + ( ψ TagBox["\[Psi]", PolyGamma] ( k - n + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) ) sinh ( 2 z ) - cosh ( 2 z ) Shi ( 4 z ) ) ( 16 z ) k ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] z 0 0 F ~ 1 ( 1 , 0 ) ( TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] ; n + 1 2 ; z TagBox["z", Hypergeometric0F1, Rule[Editable, True]] ) δ KroneckerDelta n cosh ( 2 z ) Chi ( 4 z ) - sinh ( 2 z ) Shi ( 4 z ) π - 1 2 z 1 - 2 n 4 log ( z ) I n - 1 2 ( 2 z ) - ( - 1 ) n 2 2 - 2 n ( n - 1 ) ! π z 1 2 - n ( 4 z k = 0 n 2 - 1 ( n - 1 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - 2 k + 2 n - 3 ) ! ( - cosh ( 2 z ) Chi ( 4 z ) + cosh ( 2 z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 3 2 ) ) + sinh ( 2 z ) Shi ( 4 z ) ) ( 16 z ) k + k = 0 n - 1 2 ( n - 1 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - 2 k + 2 n - 2 ) ! ( Chi ( 4 z ) sinh ( 2 z ) + ( ψ TagBox["\[Psi]", PolyGamma] ( k - n + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) ) sinh ( 2 z ) - cosh ( 2 z ) Shi ( 4 z ) ) ( 16 z ) k ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Hypergeometric0F1Regularized", 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[FractionBox[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]], SqrtBox["\[Pi]"]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], ")"]]]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["2", "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "2"]], ")"]], "!"]], " ", RowBox[List["(", 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["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["16", " ", "z"]], ")"]], "k"]]]]], "+", RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "3"]], ")"]], "!"]], " ", 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["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["16", " ", "z"]], ")"]], "k"]]]]]]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2007)

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

 2007-05-02