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 StruveL

 http://functions.wolfram.com/03.10.03.0007.01

 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[(Abs[\[Nu]] + 2 k - 1/2)!/ ((2 k)! (Abs[\[Nu]] - 2 k - 1/2)! (2 z)^(2 k)), {k, 0, Floor[(1/4) (2 Abs[\[Nu]] - 1)]}] + Cosh[(1/2) I Pi (1/2 + \[Nu]) - z] Sum[((Abs[\[Nu]] + 2 k + 1/2)! (2 z)^(-2 k - 1))/ ((2 k + 1)! (Abs[\[Nu]] - 2 k - 3/2)!), {k, 0, Floor[(1/4) (2 Abs[\[Nu]] - 3)]}]) - ((2^(1 - \[Nu]) z^(-1 + \[Nu]))/(Sqrt[Pi] (-(1/2) + \[Nu])!)) Sum[(Pochhammer[1/2, k] Pochhammer[1/2 - \[Nu], k])/(z^2/4)^k, {k, 0, \[Nu] - 1/2}] /; Element[\[Nu] - 1/2, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[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[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", RowBox[List["2", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", "k"]], "-", 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[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", RowBox[List["2", "k"]], "+", 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["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]]]]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ")"]], "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "k"]]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]

 MathML Form

 L StruveL ν ( z ) - 2 1 - ν z ν - 1 π ( ν - 1 2 ) ! k = 0 ν - 1 2 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 1 2 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( z 2 4 ) - k + - 1 z 1 2 π ( ν + 1 2 ) 2 π ( sinh ( 1 2 π ( ν + 1 2 ) - z ) k = 0 1 4 ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ) ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ) ! ( 2 k ) ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 1 2 ) ! ( 2 z ) 2 k + cosh ( 1 2 π ( ν + 1 2 ) - z ) k = 0 1 4 ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 3 ) ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 2 ) ! ( 2 z ) - 2 k - 1 ( 2 k + 1 ) ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 3 2 ) ! ) /; ν - 1 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition StruveL ν z -1 2 1 -1 ν z ν -1 1 2 ν -1 1 2 -1 k 0 ν -1 1 2 Pochhammer 1 2 k Pochhammer 1 2 -1 ν k z 2 4 -1 -1 k -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 4 2 ν -1 2 k ν -1 1 2 2 k ν -1 2 k -1 1 2 2 z 2 k -1 1 2 ν 1 2 -1 z k 0 1 4 2 ν -3 2 k ν 1 2 2 z -2 k -1 2 k 1 ν -1 2 k -1 3 2 -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[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[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", 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[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", RowBox[List["2", " ", "k"]], "+", 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["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]]]]]]]]]]], ")"]]]], SqrtBox["z"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "k"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29