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http://functions.wolfram.com/10.06.20.0017.01
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D[LerchPhi[z, s, a], {z, 2}] ==
z^(-1 + Max[0, Floor[-Re[a]]])
(LerchPhi[z, -2 + s, 1 + a + Max[0, Floor[-Re[a]]]] -
(1 + 2 a) LerchPhi[z, -1 + s, 1 + a + Max[0, Floor[-Re[a]]]] +
a (a + 1) LerchPhi[z, s, 1 + a + Max[0, Floor[-Re[a]]]]) +
Sum[((k + 1) (k + 2) z^k)/((2 + a + k)^2)^(s/2), {k, 0, Floor[-Re[a]] - 2}]
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "]"]]]]], RowBox[List["(", " ", RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["-", "2"]], "+", "s"]], ",", RowBox[List["1", "+", "a", "+", RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "]"]]]]]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["LerchPhi", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["-", "1"]], "+", "s"]], ",", RowBox[List["1", "+", "a", "+", RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "]"]]]]]], "]"]]]], "+", RowBox[List["a", RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], " ", RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", RowBox[List["1", "+", "a", "+", RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "]"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "-", "2"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], " ", SuperscriptBox["z", "k"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <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]], ",", TagBox["s", LerchPhi, Rule[Editable, True]], ",", TagBox["a", LerchPhi, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <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> <mi> a </mi> <mo> + </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True]], ",", TagBox[RowBox[List["s", "-", "2"]], LerchPhi, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", RowBox[List["max", "(", RowBox[List["0", ",", RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightFloor]"]]]], ")"]], "+", "1"]], LerchPhi, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]]] </annotation> </semantics> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True]], ",", TagBox[RowBox[List["s", "-", "1"]], LerchPhi, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", RowBox[List["max", "(", RowBox[List["0", ",", RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightFloor]"]]]], ")"]], "+", "1"]], LerchPhi, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True]], ",", TagBox["s", LerchPhi, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", RowBox[List["max", "(", RowBox[List["0", ",", RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightFloor]"]]]], ")"]], "+", "1"]], LerchPhi, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> LerchPhi </ci> <ci> z </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> LerchPhi </ci> <ci> z </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <apply> <plus /> <ci> a </ci> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </uplimit> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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