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 StruveH

 http://functions.wolfram.com/03.09.03.0007.01

 Input Form

 StruveH[\[Nu], z] == ((-1)^(\[Nu] + 1/2)/Sqrt[z]) Sqrt[2/Pi] (Sin[(1/2) Pi (\[Nu] + 1/2) + z] Sum[((-1)^k (2 k + \[Nu] - 1/2)!)/ ((2 k)! (\[Nu] - 2 k - 1/2)! (2 z)^(2 k)), {k, 0, Floor[(1/4) (2 \[Nu] - 1)]}] + Cos[(1/2) Pi (\[Nu] + 1/2) + z] Sum[((-1)^k (2 k + \[Nu] + 1/2)! (2 z)^(-2 k - 1))/ ((2 k + 1)! (\[Nu] - 2 k - 3/2)!), {k, 0, Floor[(1/4) (2 \[Nu] - 3)]}]) + (1/((\[Nu] - 1/2)! Sqrt[Pi])) (z/2)^(\[Nu] - 1) 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["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]]], SqrtBox["z"]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]]]], "+", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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["\[Nu]", "-", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]]]], "+", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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["\[Nu]", "-", RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]]]]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SqrtBox["\[Pi]"]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "-", "1"]]], 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["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], RowBox[List["-", "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]

 MathML Form

 H StruveH ν ( z ) 1 ( ν - 1 2 ) ! π ( z 2 ) ν - 1 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 + 2 π ( - 1 ) ν + 1 2 z ( sin ( 1 2 π ( ν + 1 2 ) + z ) k = 0 1 4 ( 2 ν - 1 ) ( - 1 ) k ( 2 k + ν - 1 2 ) ! ( 2 k ) ! ( - 2 k + ν - 1 2 ) ! ( 2 z ) 2 k + cos ( 1 2 π ( ν + 1 2 ) + z ) k = 0 1 4 ( 2 ν - 3 ) ( - 1 ) k ( 2 k + ν + 1 2 ) ! ( 2 z ) - 2 k - 1 ( 2 k + 1 ) ! ( - 2 k + ν - 3 2 ) ! ) /; ν - 1 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition StruveH ν z 1 ν -1 1 2 1 2 -1 z 2 -1 ν -1 k 0 ν -1 1 2 Pochhammer 1 2 k Pochhammer 1 2 -1 ν k -1 z 2 4 -1 -1 k 2 -1 1 2 -1 ν 1 2 z 1 2 -1 1 2 ν 1 2 z k 0 1 4 2 ν -1 -1 k 2 k ν -1 1 2 2 k -2 k ν -1 1 2 2 z 2 k -1 1 2 ν 1 2 z k 0 1 4 2 ν -3 -1 k 2 k ν 1 2 2 z -2 k -1 2 k 1 -2 k ν -1 3 2 -1 ν -1 1 2 [/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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]]]], "+", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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["\[Nu]", "-", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]]]], "+", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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["\[Nu]", "-", RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]]]]]]]]]]], ")"]]]], SqrtBox["z"]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", 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["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], RowBox[List["-", "k"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29