Lerch transcendent: Differentiation (formula 10.06.20.0006)

LerchPhi

$Mathematica Notation$

Zeta Functions and Polylogarithms

LerchPhi[z,s,a]

Differentiation

Low-order differentiation

With respect to a

For Phi^{^}(z,s,a)

http://functions.wolfram.com/10.06.20.0006.01

Input Form

D[LerchPhiClassical[z, s, a], {a, 2}] == s (s + 1) LerchPhiClassical[z, s + 2, a]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "2"]], "}"]]], RowBox[List["LerchPhiClassical", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]]]], "\[Equal]", RowBox[List["s", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]], RowBox[List["LerchPhiClassical", "[", RowBox[List["z", ",", RowBox[List["s", "+", "2"]], ",", "a"]], "]"]]]]]]]]

MathML Form

<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> <mover> <mi> Φ </mi> <mo> ^ </mo> </mover> <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[OverscriptBox["\[CapitalPhi]", "^"], "(", RowBox[List[TagBox["z", Rule[Editable, True]], ",", TagBox["s", Rule[Editable, True]], ",", TagBox["a", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mover> <mi> Φ </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[CapitalPhi]", "^"], "(", RowBox[List[TagBox["z", Rule[Editable, True]], ",", TagBox[RowBox[List["s", "+", "2"]], Rule[Editable, True]], ",", TagBox["a", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> a </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> <times /> <ci> s </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> LerchPhi </ci> <ci> z </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "2"]], "}"]]]]], RowBox[List["LerchPhiClassical", "[", RowBox[List["z_", ",", "s_", ",", "a_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["s", " ", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]], " ", RowBox[List["LerchPhiClassical", "[", RowBox[List["z", ",", RowBox[List["s", "+", "2"]], ",", "a"]], "]"]]]]]]]]

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

2001-10-29