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 LerchPhi

 http://functions.wolfram.com/10.06.03.0015.02

 Input Form

 LerchPhi[z, -3, a] == (2 UnitStep[-Re[a]] - 1) ((a^3 (-1 + z)^3 - 3 a^2 (-1 + z)^2 z + 3 a z (-1 + z^2) - z (1 + 4 z + z^2))/(-1 + z)^4) + (UnitStep[-Re[a]] 2 (-((1/(-1 + z)^4) (z (-1 - 3 a^2 (-1 + z)^2 + a^3 (-1 + z)^3 - 4 z - z^2 + 3 a (-1 + z^2) - 3 (-1 + z) (1 - 2 a (-1 + z) + a^2 (-1 + z)^2 + z) Ceiling[Re[a]] + 3 (-1 + a (-1 + z)) (-1 + z)^2 Ceiling[Re[a]]^2 - (-1 + z)^3 Ceiling[Re[a]]^3))) + ((a - Ceiling[Re[a]])^2)^(3/2) (1 - Ceiling[Re[a]] + Floor[Re[a]]) UnitStep[Im[a]]))/z^Ceiling[Re[a]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["z", ",", RowBox[List["-", "3"]], ",", "a"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "-", "1"]], ")"]], FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "3"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"], " ", "z"]], "+", RowBox[List["3", " ", "a", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "2", " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"]], RowBox[List["(", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "3"]]], "-", RowBox[List["4", " ", "z"]], "-", SuperscriptBox["z", "2"], "+", RowBox[List["3", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]]], "+", "z"]], ")"]], " ", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "2"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "3"]]]]], ")"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], "2"], ")"]], RowBox[List["3", "/", "2"]]], " ", 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", "]"]], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 Φ ( z , - 3 , a ) TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", "3"]], 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]] 2 ( ( ( a - Re ( a ) ) 2 ) 3 / 2 ( - Re ( a ) + Re ( a ) + 1 ) θ UnitStep ( Im ( a ) ) - 1 ( z - 1 ) 4 ( z ( a 3 ( z - 1 ) 3 - Re ( a ) 3 ( z - 1 ) 3 - 3 a 2 ( z - 1 ) 2 + 3 ( a ( z - 1 ) - 1 ) Re ( a ) 2 ( z - 1 ) 2 - 3 ( a 2 ( z - 1 ) 2 - 2 a ( z - 1 ) + z + 1 ) Re ( a ) ( z - 1 ) - z 2 - 4 z + 3 a ( z 2 - 1 ) - 1 ) ) ) θ UnitStep ( - Re ( a ) ) z - Re ( a ) + ( 2 θ UnitStep ( - Re ( a ) ) - 1 ) ( a 3 ( z - 1 ) 3 - 3 a 2 z ( z - 1 ) 2 + 3 a z ( z 2 - 1 ) - z ( z 2 + 4 z + 1 ) ) ( z - 1 ) 4 LerchPhi z -3 a 2 a -1 a 2 3 2 -1 a a 1 UnitStep a -1 1 z -1 4 -1 z a 3 z -1 3 -1 a 3 z -1 3 -1 3 a 2 z -1 2 3 a z -1 -1 a 2 z -1 2 -1 3 a 2 z -1 2 -1 2 a z -1 z 1 a z -1 -1 z 2 -1 4 z 3 a z 2 -1 -1 UnitStep -1 a z -1 a 2 UnitStep -1 a -1 a 3 z -1 3 -1 3 a 2 z z -1 2 3 a z z 2 -1 -1 z z 2 4 z 1 z -1 4 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LerchPhi", "[", RowBox[List["z_", ",", RowBox[List["-", "3"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "3"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"], " ", "z"]], "+", RowBox[List["3", " ", "a", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "z"]], "+", SuperscriptBox["z", "2"]]], ")"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], " ", "2", " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "3"]]], "-", RowBox[List["4", " ", "z"]], "-", SuperscriptBox["z", "2"], "+", RowBox[List["3", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"]]], "+", "z"]], ")"]], " ", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "2"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "3"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], "2"], ")"]], RowBox[List["3", "/", "2"]]], " ", 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", "]"]], "]"]]]]]], ")"]]]]]]]]]]

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

 2001-10-29