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; }

 StruveL

 http://functions.wolfram.com/03.10.20.0012.01

 Input Form

 D[StruveL[\[Nu], z], {z, \[Alpha]}] == Sum[((-1)^\[Nu] 2^(-1 - 2 k - \[Nu]) z^(1 + 2 k - \[Alpha] + \[Nu]) (Log[z] + PolyGamma[-1 - 2 k - \[Nu]] - PolyGamma[2 + 2 k - \[Alpha] + \[Nu]]))/((-2 - 2 k - \[Nu])! Gamma[3/2 + k] Gamma[3/2 + k + \[Nu]] Gamma[2 + 2 k - \[Alpha] + \[Nu]]), {k, 0, -Floor[(\[Nu] + 3)/2]}] + (2^(\[Alpha] - 2 (1 + \[Nu]) + 4 Floor[(1 + \[Nu])/2]) Sqrt[Pi] z^(1 - \[Alpha] + \[Nu] - 2 Floor[(1 + \[Nu])/2]) Gamma[2 + \[Nu] - 2 Floor[(1 + \[Nu])/2]] HypergeometricPFQRegularized[ {1, (1/2) (2 + \[Nu] - 2 Floor[(1 + \[Nu])/2]), (1/2) (3 + \[Nu] - 2 Floor[(1 + \[Nu])/2])}, {3/2 - Floor[(1 + \[Nu])/2], (1/2) (2 - \[Alpha] + \[Nu] - 2 Floor[(1 + \[Nu])/2]), (1/2) (3 - \[Alpha] + \[Nu] - 2 Floor[(1 + \[Nu])/2]), 3/2 + \[Nu] - Floor[(1 + \[Nu])/2]}, z^2/4])/(-1)^Floor[(1 + \[Nu])/2] /; Element[-\[Nu], Integers] && -\[Nu] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["-", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], "]"]]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]], " ", SuperscriptBox["2", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Alpha]", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]", "-", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]]]]]

 MathML Form

 α L StruveL ν ( z ) z α ( - 1 ) - ν + 1 2 2 α - 2 ( ν + 1 ) + 4 ν + 1 2 π Γ ( ν - 2 ν + 1 2 + 2 ) 3 F ~ 4 ( 1 , 1 2 ( ν - 2 ν + 1 2 + 2 ) , 1 2 ( ν - 2 ν + 1 2 + 3 ) ; 3 2 - ν + 1 2 , 1 2 ( ν - α - 2 ν + 1 2 + 2 ) , 1 2 ( ν - α - 2 ν + 1 2 + 3 ) , ν - ν + 1 2 + 3 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "\[RightFloor]"]]]], "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "\[RightFloor]"]]]], "+", "3"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["3", "2"], "-", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "\[RightFloor]"]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Alpha]", "-", RowBox[List["2", " ", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "\[RightFloor]"]]]], "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Alpha]", "-", RowBox[List["2", " ", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "\[RightFloor]"]]]], "+", "3"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "\[RightFloor]"]], "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] z ν - α - 2 ν + 1 2 + 1 + k = 0 - ν + 3 2 ( - 1 ) ν 2 - 2 k - ν - 1 z 2 k - α + ν + 1 ( log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( - 2 k - ν - 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 k - α + ν + 2 ) ) ( - 2 k - ν - 2 ) ! Γ ( k + 3 2 ) Γ ( k + ν + 3 2 ) Γ ( 2 k - α + ν + 2 ) /; - ν TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + Condition z α StruveL ν z -1 -1 ν 1 2 -1 2 α -1 2 ν 1 4 ν 1 2 -1 1 2 Gamma ν -1 2 ν 1 2 -1 2 HypergeometricPFQRegularized 1 1 2 ν -1 2 ν 1 2 -1 2 1 2 ν -1 2 ν 1 2 -1 3 3 2 -1 ν 1 2 -1 1 2 ν -1 α -1 2 ν 1 2 -1 2 1 2 ν -1 α -1 2 ν 1 2 -1 3 ν -1 ν 1 2 -1 3 2 z 2 4 -1 z ν -1 α -1 2 ν 1 2 -1 1 k 0 -1 ν 3 2 -1 -1 ν 2 -2 k -1 ν -1 z 2 k -1 α ν 1 z PolyGamma -2 k -1 ν -1 -1 PolyGamma 2 k -1 α ν 2 -2 k -1 ν -2 Gamma k 3 2 Gamma k ν 3 2 Gamma 2 k -1 α ν 2 -1 -1 ν SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["StruveL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["-", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], "]"]]]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]], " ", SuperscriptBox["2", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Alpha]", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "+", "\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]", "-", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]]]]]]]

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

 2001-10-29