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LerchPhi






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > LerchPhi[z,s,a] > Differentiation > Low-order differentiation > With respect to s > For Phi(z,s,a)





http://functions.wolfram.com/10.06.20.0024.01









  


  










Input Form





D[LerchPhi[z, s, a], s] == D[LerchPhi[z, s, a - Floor[Re[a]]], s]/ z^Floor[Re[a]] + ((Sign[Re[a]]/2) Sum[Log[(1/2 + ((1 - 2 a)/2) Sign[Re[a]] + k)^2]/ (z^(Sign[Re[a]] k) ((1/2 + ((1 - 2 a)/2) Sign[Re[a]] + k)^2)^(s/2)), {k, 0, Abs[Floor[Re[a]]] - 1}])/z^UnitStep[Re[a]] /; !(Element[a, Integers] && a > 0)










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["s_"]]], RowBox[List["LerchPhi", "[", RowBox[List["z_", ",", "s_", ",", "a_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["s"]]], RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", RowBox[List["a", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], "+", "k"]], ")"]], "2"], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], " ", "k"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["a", "\[Element]", "Integers"]], "&&", RowBox[List["a", ">", "0"]]]], ")"]]]]]]]]]]










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





2007-05-02





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