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http://functions.wolfram.com/10.06.06.0034.01
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LerchPhi[z, s, a] \[Proportional] LerchPhi[z, s, Subscript[a, 0]] +
s (Zeta[1 + s, Subscript[a, 0]] -
2 Zeta[1 + s, Subscript[a, 0] + Max[Floor[-Re[Subscript[a, 0]]] + 1,
0]]) (a - Subscript[a, 0]) + ((s (s + 1))/2)
(Sum[z^k/((Subscript[a, 0] + k)^2 ((Subscript[a, 0] + k)^2)^(s/2)),
{k, 0, Floor[-Re[Subscript[a, 0]]]}] +
z^Max[Floor[-Re[Subscript[a, 0]]] + 1, 0] LerchPhi[z, 2 + s,
Subscript[a, 0] + Max[Floor[-Re[Subscript[a, 0]]] + 1, 0]])
(a - Subscript[a, 0])^2 + \[Ellipsis] /; (a -> Subscript[a, 0])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", SubscriptBox["a", "0"]]], "]"]], "+", RowBox[List["s", " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "s"]], ",", SubscriptBox["a", "0"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "s"]], ",", RowBox[List[SubscriptBox["a", "0"], "+", RowBox[List["Max", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["a", "0"], "]"]]]], "]"]], "+", "1"]], ",", "0"]], "]"]]]]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "0"]]], ")"]]]], " ", "+", RowBox[List[FractionBox[RowBox[List["s", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]]]], "2"], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["a", "0"], "]"]]]], "]"]]], FractionBox[SuperscriptBox["z", "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "0"], "+", "k"]], ")"]], "2"], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "0"], "+", "k"]], ")"]], "2"], ")"]], FractionBox["s", "2"]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["Max", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["a", "0"], "]"]]]], "]"]], "+", "1"]], ",", "0"]], "]"]]], RowBox[List["LerchPhi", "[", RowBox[List["z", ",", RowBox[List["2", "+", "s"]], ",", RowBox[List[SubscriptBox["a", "0"], "+", RowBox[List["Max", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["a", "0"], "]"]]]], "]"]], "+", "1"]], ",", "0"]], "]"]]]]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["a", "\[Rule]", SubscriptBox["a", "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> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["s", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["s", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["a", "0"], LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> + </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["s", "+", "1"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["a", "0"], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["s", "+", "1"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["max", "(", RowBox[List[RowBox[List[RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", SubscriptBox["a", "0"], ")"]]]], "\[RightFloor]"]], "+", "1"]], ",", "0"]], ")"]], "+", SubscriptBox["a", "0"]]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["s", "+", "2"]], LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["max", "(", RowBox[List[RowBox[List[RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", SubscriptBox["a", "0"], ")"]]]], "\[RightFloor]"]], "+", "1"]], ",", "0"]], ")"]], "+", SubscriptBox["a", "0"]]], LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </munderover> <mfrac> <msup> <mi> z </mi> <mi> k </mi> </msup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <ci> s </ci> <apply> <plus /> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <max /> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> s </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <max /> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> LerchPhi </ci> <ci> z </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <max /> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> a </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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