html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 StruveH

 http://functions.wolfram.com/03.09.06.0019.01

 Input Form

 StruveH[\[Nu], z] == (1/Subscript[z, 0])^(\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Subscript[z, 0]^(\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Sum[((Subscript[z, 0]^(1 - k + \[Nu])/(2^\[Nu] (Sqrt[Pi] k! Gamma[1/2 + \[Nu]]))) Sum[(-1)^(m + i) Binomial[i, m] Pochhammer[-\[Nu], i - m] Sum[(((-1)^(p - 1) 2^(2 p - m) Pochhammer[-m, 2 (m - p)] Pochhammer[\[Nu], p])/(m - p)!) Sum[((p - j - 1)! Pochhammer[2 j + \[Nu] - k + 2, k - i - 1] Subscript[z, 0]^(2 j))/(2^(2 j) (j! (p - 2 j - 1)! Pochhammer[1 - p - \[Nu], j] Pochhammer[\[Nu], j + 1])), {j, 0, p - 1}], {p, 0, m}], {i, 1, k - 1}, {m, 0, i}] + (1/(Subscript[z, 0]^k k!)) Sum[(-1)^(m + k) Binomial[k, m] Pochhammer[-\[Nu], k - m] Sum[(((-1)^(p - 1) 2^(2 p - m) Pochhammer[-m, 2 (m - p)] Pochhammer[\[Nu], p])/(m - p)!) ((Subscript[z, 0]/2) Sum[((p - j - 1)!/(j! (p - 2 j - 1)! Pochhammer[1 - p - \[Nu], j] Pochhammer[\[Nu], j + 1])) (Subscript[z, 0]^2/4)^j StruveH[\[Nu] - 1, Subscript[z, 0]], {j, 0, p - 1}] - Sum[((p - j)!/(j! (p - 2 j)! Pochhammer[1 - p - \[Nu], j] Pochhammer[\[Nu], j])) (Subscript[z, 0]^2/4)^j StruveH[\[Nu], Subscript[z, 0]], {j, 0, p}]), {p, 0, m}], {m, 0, k}]) (z - Subscript[z, 0])^k, {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SubsuperscriptBox["z", "0", RowBox[List["1", "-", "k", "+", "\[Nu]"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "i"]]], " ", RowBox[List["Binomial", "[", RowBox[List["i", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["i", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "p"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "p"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "j"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "+", "\[Nu]", "-", "k", "+", "2"]], ",", RowBox[List["k", "-", "i", "-", "1"]]]], "]"]], " ", SubsuperscriptBox["z", "0", RowBox[List["2", "j"]]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "p", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]]]]]]]]]]]]]]]], "+", RowBox[List[FractionBox[SubsuperscriptBox["z", "0", RowBox[List["-", "k"]]], RowBox[List["k", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "p"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "p"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SubscriptBox["z", "0"], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "p", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubsuperscriptBox["z", "0", "2"], "4"], ")"]], "j"], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", SubscriptBox["z", "0"]]], "]"]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "p"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "p", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubsuperscriptBox["z", "0", "2"], "4"], ")"]], "j"], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", SubscriptBox["z", "0"]]], "]"]]]]]]]], ")"]]]]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]

 MathML Form

 H StruveH ν ( z ) ( 1 z 0 ) ν arg ( z - z 0 ) 2 π z 0 ν arg ( z - z 0 ) 2 π k = 0 ( z 0 - k k ! m = 0 k ( - 1 ) k + m ( k m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["m", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ν ) k - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] p = 0 m ( - 1 ) p - 1 2 2 p - m ( - m ) 2 ( m - p ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]], Pochhammer] ( ν ) p TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "p"], Pochhammer] ( m - p ) ! ( 1 2 z 0 j = 0 p - 1 ( - j + p - 1 ) ! j ! ( - 2 j + p - 1 ) ! ( - p - ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "-", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] ( ν ) j + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] ( z 0 2 4 ) j H StruveH ν - 1 ( z 0 ) - j = 0 p ( p - j ) ! j ! ( p - 2 j ) ! ( - p - ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "-", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] ( ν ) j TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "j"], Pochhammer] ( z 0 2 4 ) j H StruveH ν ( z 0 ) ) + 2 - ν z 0 - k + ν + 1 π k ! Γ ( ν + 1 2 ) i = 1 k - 1 m = 0 i ( - 1 ) i + m ( i m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["m", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ν ) i - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["i", "-", "m"]]], Pochhammer] p = 0 m ( - 1 ) p - 1 2 2 p - m ( - m ) 2 ( m - p ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]], Pochhammer] ( ν ) p TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "p"], Pochhammer] ( m - p ) ! j = 0 p - 1 2 - 2 j ( - j + p - 1 ) ! ( 2 j - k + ν + 2 ) - i + k - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k", "+", "\[Nu]", "+", "2"]], ")"]], RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "1"]]], Pochhammer] z 0 2 j j ! ( - 2 j + p - 1 ) ! ( - p - ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "-", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] ( ν ) j + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] ) ( z - z 0 ) k StruveH ν z 1 Subscript z 0 -1 ν z -1 Subscript z 0 2 -1 Subscript z 0 ν z -1 Subscript z 0 2 -1 k 0 Subscript z 0 -1 k k -1 m 0 k -1 k m Binomial k m Pochhammer -1 ν k -1 m p 0 m -1 p -1 2 2 p -1 m Pochhammer -1 m 2 m -1 p Pochhammer ν p m -1 p -1 1 2 Subscript z 0 j 0 p -1 -1 j p -1 j -2 j p -1 Pochhammer -1 p -1 ν 1 j Pochhammer ν j 1 -1 Subscript z 0 2 4 -1 j StruveH ν -1 Subscript z 0 -1 j 0 p p -1 j j p -1 2 j Pochhammer -1 p -1 ν 1 j Pochhammer ν j -1 Subscript z 0 2 4 -1 j StruveH ν Subscript z 0 2 -1 ν Subscript z 0 -1 k ν 1 1 2 k Gamma ν 1 2 -1 m 0 i i 1 k -1 -1 i m Binomial i m Pochhammer -1 ν i -1 m p 0 m -1 p -1 2 2 p -1 m Pochhammer -1 m 2 m -1 p Pochhammer ν p m -1 p -1 j 0 p -1 2 -2 j -1 j p -1 Pochhammer 2 j -1 k ν 2 -1 i k -1 Subscript z 0 2 j j -2 j p -1 Pochhammer -1 p -1 ν 1 j Pochhammer ν j 1 -1 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["zz", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["1", "-", "k", "+", "\[Nu]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "i"]]], " ", RowBox[List["Binomial", "[", RowBox[List["i", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["i", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "p"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", 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RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[SubsuperscriptBox["zz", "0", RowBox[List["-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "p"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", 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RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "p"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubsuperscriptBox["zz", "0", "2"], "4"], ")"]], "j"], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "p", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]], "!"]]]]]]]]]]], RowBox[List["k", "!"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]]]]]]

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

 2007-05-02