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 StruveL

 http://functions.wolfram.com/03.10.03.0006.01

 Input Form

 StruveL[\[Nu], z] == (-(1/Sqrt[z])) E^((1/2) Pi I (1/2 + \[Nu])) Sqrt[2/Pi] (Sinh[(1/2) I Pi (1/2 + \[Nu]) - z] Sum[(2 k - \[Nu] - 1/2)!/ ((2 k)! (-2 k - \[Nu] - 1/2)! (2 z)^(2 k)), {k, 0, Floor[(-(1/4)) (2 \[Nu] + 1)]}] + Cosh[(1/2) I Pi (1/2 + \[Nu]) - z] Sum[((2 k - \[Nu] + 1/2)! (2 z)^(-2 k - 1))/ ((2 k + 1)! (-2 k - \[Nu] - 3/2)!), {k, 0, Floor[(-(1/4)) (2 \[Nu] + 3)]}]) /; Element[-\[Nu] - 1/2, Integers] && -\[Nu] - 1/2 >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", SqrtBox["z"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]]]], "-", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]], "+", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]]]], "-", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "\[Nu]", "-", FractionBox["3", "2"]]], ")"]], "!"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 L StruveL ν ( z ) - 1 z 1 2 π ( ν + 1 2 ) 2 π ( sinh ( 1 2 π ( ν + 1 2 ) - z ) k = 0 - 1 4 ( 2 ν + 1 ) ( 2 k - ν - 1 2 ) ! ( 2 k ) ! ( - 2 k - ν - 1 2 ) ! ( 2 z ) 2 k + cosh ( 1 2 π ( ν + 1 2 ) - z ) k = 0 - 1 4 ( 2 ν + 3 ) ( 2 k - ν + 1 2 ) ! ( 2 z ) - 2 k - 1 ( 2 k + 1 ) ! ( - 2 k - ν - 3 2 ) ! ) /; - ν - 1 2 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition StruveL ν z -1 1 z 1 2 -1 1 2 ν 1 2 2 -1 1 2 1 2 ν 1 2 -1 z k 0 -1 1 4 2 ν 1 2 k -1 ν -1 1 2 2 k -2 k -1 ν -1 1 2 2 z 2 k -1 1 2 ν 1 2 -1 z k 0 -1 1 4 2 ν 3 2 k -1 ν 1 2 2 z -2 k -1 2 k 1 -2 k -1 ν -1 3 2 -1 -1 ν -1 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]]]], "-", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]], "+", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]]]], "-", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "\[Nu]", "-", FractionBox["3", "2"]]], ")"]], "!"]]]]]]]]]]], ")"]]]], SqrtBox["z"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29