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 StruveL

 http://functions.wolfram.com/03.10.20.0017.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["StruveL", 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[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["CoshIntegral", "[", "z", "]"]]]], "-", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["SinhIntegral", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], " ", RowBox[List["BesselI", "[", 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[RowBox[List["\[Pi]", " ", "z"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", "z"]]], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "p"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "p"]]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["n", "-", "k", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "p"]]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]], "-", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["BesselI", "[", 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