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 StruveL

 http://functions.wolfram.com/03.10.06.0039.01

 Input Form

 StruveL[\[Nu], z] \[Proportional] (1/Sqrt[2 Pi z]) (E^z (Sum[((Pochhammer[1/2 + \[Nu], k] Pochhammer[1/2 - \[Nu], k])/k!) (1/(2 z))^k, {k, 0, n}] + O[1/z^(n + 1)]) + I E^(-z - I Pi \[Nu]) (Sum[((Pochhammer[1/2 + \[Nu], k] Pochhammer[1/2 - \[Nu], k])/k!) (-(1/(2 z)))^k, {k, 0, n}] + O[1/z^(n + 1)])) - ((2^(1 - \[Nu]) z^(\[Nu] - 1))/(Sqrt[Pi] Gamma[1/2 + \[Nu]])) (Sum[Pochhammer[1/2, k] Pochhammer[1/2 - \[Nu], k] (4/z^2)^k, {k, 0, n}] + O[1/z^(2 n + 2)]) /; -Pi < Arg[z] < Pi/2 && (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["2", "\[Pi]", " ", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["2", " ", "z"]]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "z"]]]]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]]]], RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], 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["4", SuperscriptBox["z", "2"]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", "n"]], "+", "2"]]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

 MathML Form

 L StruveL ν ( z ) 1 2 π z ( z ( k = 0 n ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 1 2 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( 1 2 z ) k + O ( 1 z n + 1 ) ) + - z - π ν ( k = 0 n ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 1 2 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( - 1 2 z ) k + O ( 1 z n + 1 ) ) ) - 2 1 - ν z ν - 1 π Γ ( ν + 1 2 ) ( k = 0 n ( 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] ( 4 z 2 ) k + O ( 1 z 2 n + 2 ) ) /; - π < arg ( z ) < π 2 ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional StruveL ν z 1 2 z 1 2 -1 z k 0 n Pochhammer ν 1 2 k Pochhammer 1 2 -1 ν k k -1 1 2 z -1 k O 1 z n 1 -1 -1 z -1 ν k 0 n Pochhammer ν 1 2 k Pochhammer 1 2 -1 ν k k -1 -1 1 2 z -1 k O 1 z n 1 -1 -1 2 1 -1 ν z ν -1 1 2 Gamma ν 1 2 -1 k 0 n Pochhammer 1 2 k Pochhammer 1 2 -1 ν k 4 z 2 -1 k O 1 z 2 n 2 -1 -1 z 2 -1 Rule z [/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[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["2", " ", "z"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "z"]]]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]", " ", "z"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], 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["4", SuperscriptBox["z", "2"]], ")"]], "k"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]]]], "]"]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

 2007-05-02