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http://functions.wolfram.com/03.10.06.0025.01
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StruveL[\[Nu], z] == Exp[2 Pi I \[Nu] Floor[Arg[z - x]/(2 Pi)]]
Sum[((x^(1 - k + \[Nu])/(2^\[Nu] (Sqrt[Pi] Gamma[1/2 + \[Nu]] k!)))
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[((-1)^j (p - j - 1)! Pochhammer[2 j + \[Nu] - k + 2,
k - i - 1] z^(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/(x^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)!)
((x/2) Sum[((p - j - 1)!/(j! (p - 2 j - 1)! Pochhammer[
1 - p - \[Nu], j] Pochhammer[\[Nu], j + 1])) (-(x^2/4))^
j StruveL[\[Nu] - 1, x], {j, 0, p - 1}] -
Sum[((p - j)!/(j! (p - 2 j)! Pochhammer[1 - p - \[Nu], j]
Pochhammer[\[Nu], j])) (-(x^2/4))^j StruveL[\[Nu], x],
{j, 0, p}]), {p, 0, m}], {m, 0, k}]) (z - x)^k,
{k, 0, Infinity}] /; Element[x, Reals] && x < 0
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <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> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> 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]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mi> p </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "p"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "-", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "-", "\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <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> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> i </mi> <mo> + </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> i </mi> </mtd> </mtr> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> 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]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> i </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["i", "-", "m"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "p"]], ")"]]]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( 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</apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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