|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/03.10.20.0012.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> α </mi> </msup> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </ci> </annotation-xml> </semantics> </mstyle> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> α </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> α </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 4 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> ν </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> α </ci> </degree> </bvar> <apply> <ci> StruveL </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </list> <list> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <ci> z </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <ci> ν </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| 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)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
