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 PolyLog

 http://functions.wolfram.com/10.08.06.0036.01

 Input Form

 PolyLog[n, z] == (-((z - 1)^(n - 1)/(n - 1)!)) Sum[Subscript[p, n - 1, k] (z - 1)^k (-EulerGamma - PolyGamma[n] + Log[Log[z]/(z - 1)]), {k, 0, Infinity}] + Zeta[n] + Sum[((Zeta[n - j] (z - 1)^j)/j!) Sum[Subscript[p, j, k] (z - 1)^k, {k, 0, Infinity}], {j, 1, n - 2}] + Sum[((Zeta[n - j] (z - 1)^j)/j!) Sum[Subscript[p, j, k] (z - 1)^k, {k, 0, Infinity}], {j, n, Infinity}] - ((z - 1)^(n - 1)/(n - 1)!) (I Pi + 2 I Pi Floor[-(Arg[z - 1]/(2 Pi)) - (1/(2 Pi)) Arg[Log[z]/(z - 1)]] + Log[z - 1]) Sum[Subscript[p, n - 1, k] (z - 1)^k, {k, 0, Infinity}] /; Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(((-1)^i (j i - k + i))/(i + 1)) Subscript[p, j, k - i], {i, 1, k}] && Element[k, Integers] && k > 0 && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"], RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List["Log", "[", FractionBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["z", "-", "1"]]], "]"]]]], ")"]]]]]]]], "+", RowBox[List["Zeta", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "2"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "j"]]]]], RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "n"]], "\[Infinity]"], " ", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "j"]]]]], RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List["Arg", "[", FractionBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["z", "-", "1"]]], "]"]]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[SubscriptBox["p", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], RowBox[List["(", RowBox[List[RowBox[List["j", " ", "i"]], "-", "k", "+", "i"]], ")"]]]], RowBox[List["i", "+", "1"]]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "i"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 Li PolyLog n ( z ) - ( z - 1 ) n - 1 ( n - 1 ) ! ( π + 2 π - 1 2 π arg ( log ( z ) z - 1 ) - arg ( z - 1 ) 2 π + log ( z - 1 ) ) k = 0 p n - 1 , k ( z - 1 ) k - ( z - 1 ) n - 1 ( n - 1 ) ! k = 0 p n - 1 , k ( z - 1 ) k ( log ( log ( z ) z - 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( n ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + j = 1 n - 2 ζ ( n - j ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["n", "-", "j"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] ( z - 1 ) j j ! k = 0 p j , k ( z - 1 ) k + j = n ζ ( n - j ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["n", "-", "j"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] ( z - 1 ) j j ! k = 0 p j , k ( z - 1 ) k + ζ ( n ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["n", Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] /; p j , 0 1 p j , k 1 k i = 1 k ( - 1 ) i ( j i + i - k ) i + 1 p j , k - i k TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyLog n z -1 z -1 n -1 n -1 -1 2 -1 1 2 -1 z z -1 -1 -1 z -1 2 -1 z -1 k 0 Subscript p n -1 k z -1 k -1 z -1 n -1 n -1 -1 k 0 Subscript p n -1 k z -1 k z z -1 -1 -1 PolyGamma n -1 j 1 n -2 Zeta n -1 j z -1 j j -1 k 0 Subscript p j k z -1 k j n Zeta n -1 j z -1 j j -1 k 0 Subscript p j k z -1 k Zeta n Subscript p j 0 1 Subscript p j k 1 k -1 i 1 k -1 i j i i -1 k i 1 -1 Subscript p j k -1 i k SuperPlus n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List["Log", "[", FractionBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["z", "-", "1"]]], "]"]]]], ")"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "+", RowBox[List["Zeta", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "j"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]], RowBox[List["j", "!"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "n"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "j"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]], RowBox[List["j", "!"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["z", "-", "1"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["p", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", RowBox[List["(", RowBox[List[RowBox[List["j", " ", "i"]], "-", "k", "+", "i"]], ")"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "i"]]]]]]], RowBox[List["i", "+", "1"]]]]], "k"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02

© 1998-2013 Wolfram Research, Inc.