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 StruveH

 http://functions.wolfram.com/03.09.06.0064.01

 Input Form

 StruveH[\[Nu], z] \[Proportional] ((1/(2 Sqrt[2 Pi])) z^(\[Nu] + 1) ((((1 - Sqrt[z^2]/z)/E^(I z) + E^(I z) (1 + Sqrt[z^2]/z)) (Sum[((Pochhammer[1/2 + \[Nu], k] Pochhammer[1/2 - \[Nu], k])/k!) (-(I/(2 Sqrt[z^2])))^k, {k, 0, n}] + O[1/z^(n + 1)]))/ E^(((3 + 2 \[Nu])/4) I Pi) + E^(((3 + 2 \[Nu])/4) I Pi) ((1 + Sqrt[z^2]/z)/E^(I z) + E^(I z) (1 - Sqrt[z^2]/z)) (Sum[((Pochhammer[1/2 + \[Nu], k] Pochhammer[1/2 - \[Nu], k])/k!) (I/(2 Sqrt[z^2]))^k, {k, 0, n}] + O[1/z^(n + 1)])))/ (z^2)^((3 + 2 \[Nu])/4) + ((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)]) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox[RowBox[List["2", "\[Pi]"]]]]]], SuperscriptBox["z", RowBox[List["\[Nu]", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["-", FractionBox[RowBox[List["3", "+", RowBox[List["2", "\[Nu]"]]]], "4"]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " "]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "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["(", RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]]]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " "]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "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["\[ImaginaryI]", RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]], ")"]], "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["(", 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["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 H StruveH ν ( z ) 1 2 2 π z ν + 1 ( z 2 ) - 3 + 2 ν 4 ( 3 + 2 ν 4 π ( - z ( z 2 z + 1 ) + z ( 1 - z 2 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 ! ( 2 z 2 ) k + O ( 1 z n + 1 ) ) + - 3 + 2 ν 4 π ( z ( z 2 z + 1 ) + - z ( 1 - z 2 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 ! ( - 2 z 2 ) 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 ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional StruveH ν z 1 2 2 1 2 -1 z ν 1 z 2 -1 3 2 ν 4 -1 3 2 ν 4 -1 -1 z z 2 1 2 z -1 1 z 1 -1 z 2 1 2 z -1 k 0 n Pochhammer ν 1 2 k Pochhammer 1 2 -1 ν k k -1 2 z 2 1 2 -1 k O 1 z n 1 -1 -1 3 2 ν 4 -1 z z 2 1 2 z -1 1 -1 z 1 -1 z 2 1 2 z -1 k 0 n Pochhammer ν 1 2 k Pochhammer 1 2 -1 ν k k -1 -1 2 z 2 1 2 -1 k O 1 z n 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 -1 4 z 2 -1 k O 1 z 2 n 2 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["\[Nu]", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "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["(", RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "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["\[ImaginaryI]", RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "+", 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["(", 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["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02