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 StruveH

 http://functions.wolfram.com/03.09.20.0017.01

 Input Form

 Derivative[1, 0][StruveH][-n - 1/2, z] == (-1)^n (2 CosIntegral[z] - CosIntegral[2 z]) BesselJ[n + 1/2, z] + (SinIntegral[2 z] - 2 SinIntegral[z]) BesselJ[-n - 1/2, z] - (((-1)^n n!)/2) Sum[(1/(k! (n - k))) (z/2)^(k - n) BesselJ[k + 1/2, z], {k, 0, n - 1}] + ((n! Sqrt[Pi])/2) Sum[(2^k/((n - k)! k)) Sum[(z^(p - k + 1/2)/p!) ((-1)^n (BesselJ[p - 1/2, 2 z] - 2^(1/2 - p) BesselJ[p - 1/2, z]) BesselJ[n - k + 1/2, z] - (-1)^(k + p) (BesselJ[1/2 - p, 2 z] - 2^(1/2 - p) BesselJ[1/2 - p, z]) BesselJ[k - n - 1/2, z]), {p, 0, k - 1}], {k, 1, n}] /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 H StruveH - n - 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) ( - 1 ) n J n + 1 2 ( z ) ( 2 Ci ( z ) - Ci ( 2 z ) ) + J - n - 1 2 ( z ) ( Si ( 2 z ) - 2 Si ( z ) ) - 1 2 ( ( - 1 ) n n ! ) k = 0 n - 1 1 k ! ( n - k ) ( z 2 ) k - n J k + 1 2 ( z ) + n ! π 2 k = 1 n 2 k ( n - k ) ! k p = 0 k - 1 z p - k + 1 2 p ! ( ( - 1 ) n ( J p - 1 2 ( 2 z ) - 2 1 2 - p J p - 1 2 ( z ) ) J n - k + 1 2 ( z ) - ( - 1 ) k + p ( J 1 2 - p ( 2 z ) - 2 1 2 - p J 1 2 - p ( z ) ) J k - n - 1 2 ( z ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript StruveH -1 n -1 1 2 z -1 n BesselJ n 1 2 z 2 CosIntegral z -1 CosIntegral 2 z BesselJ -1 n -1 1 2 z SinIntegral 2 z -1 2 SinIntegral z -1 1 2 -1 n n k 0 n -1 1 k n -1 k -1 z 2 -1 k -1 n BesselJ k 1 2 z n 1 2 2 -1 k 1 n 2 k n -1 k k -1 p 0 k -1 z p -1 k 1 2 p -1 -1 n BesselJ p -1 1 2 2 z -1 2 1 2 -1 p BesselJ p -1 1 2 z BesselJ n -1 k 1 2 z -1 -1 k p BesselJ 1 2 -1 p 2 z -1 2 1 2 -1 p BesselJ 1 2 -1 p z BesselJ k -1 n -1 1 2 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["StruveH", 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[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["CosIntegral", "[", "z", "]"]]]], "-", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["SinIntegral", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["k", "-", "n"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["p", "-", "k", "+", FractionBox["1", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", RowBox[List["2", " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "p"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "-", "k", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "p"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["2", " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "p"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["k", "-", "n", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]], ")"]]]], RowBox[List["p", "!"]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", "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