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LerchPhi






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > LerchPhi[z,s,a] > Specific values > Specialized values > For fixed z, a > For Phi(z,s,a)





http://functions.wolfram.com/10.06.03.0061.01









  


  










Input Form





LerchPhi[z, 3, a] == (1/a^3) (HypergeometricPFQ[{1, a, a, a}, {1 + a, 1 + a, 1 + a}, z] (1 - 2 UnitStep[-Re[a]])) + (2 ((z HypergeometricPFQ[{1, 1 + a - Ceiling[Re[a]], 1 + a - Ceiling[Re[a]], 1 + a - Ceiling[Re[a]]}, {2 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]]}, z])/(1 + a - Ceiling[Re[a]])^3 + ((1 - Ceiling[Re[a]] + Floor[Re[a]]) UnitStep[Im[a]])/ ((a - Ceiling[Re[a]])^2)^(3/2)) UnitStep[-Re[a]])/z^Ceiling[Re[a]]










Standard Form





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MathML Form







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<ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LerchPhi", "[", RowBox[List["z_", ",", "3", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "a", ",", "a", ",", "a"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]], ")"]]]], SuperscriptBox["a", "3"]], "+", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", RowBox[List["2", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", RowBox[List["2", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]]], "}"]], ",", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], "3"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "a", "]"]], "]"]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], "2"], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]]]]]]










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





2007-05-02