defined in the
complex plane
over the open
unit disk
. Its definition on the whole
complex plane
then follows uniquely via
analytic continuation
.
Note that the similar
notation
is used for the
logarithmic
integral
.
The polylogarithm is also denoted
and equal to
where
is the
Lerch transcendent
(Erdélyi
et
al.
1981, p. 30). The polylogarithm arises in Feynman diagram integrals
(and, in particular, in the computation of quantum electrodynamics corrections to
the electrons gyromagnetic ratio), and the special cases
and
are called the
dilogarithm
and
trilogarithm
, respectively. The polylogarithm
is implemented in the
Wolfram Language
as
PolyLog
[
n
,
The polylogarithm also arises in the closed form of the integrals of the
Fermi-Dirac
distribution
that the meaning of
for fixed complex
is not completely well-defined, since it depends on how
is approached in four-dimensional
-space.
The polylogarithm of
negative integer
order arises
in sums
of the form
No similar formulas of this type are known for higher orders (Lewin 1991, p. 2).
appears in the third-order correction
term in the gyromagnetic ratio of the electron.
The derivative of a polylogarithm is itself a polylogarithm,
A number of remarkable identities exist for polylogarithms, including the amazing identity satisfied by
,
where
(OEIS
A073011
) is the smallest
Salem
constant
, i.e., the largest positive root of the polynomial in
Lehmer's
Mahler measure problem
(Cohen
et al.
1992; Bailey and Broadhurst 1999;
Borwein and Bailey 2003, pp. 8-9).
No general
algorithm
is known for integration of polylogarithms
of functions.
Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants."
Math.
Comput.
66
, 903-913, 1997.
Bailey, D. H. and Broadhurst,
D. J. "A Seventeenth-Order Polylogarithm Ladder." 20 Jun 1999.
http://arxiv.org/abs/math.CA/9906134
.
Borwein,
J. and Bailey, D.
Mathematics
by Experiment: Plausible Reasoning in the 21st Century.
Wellesley, MA: A
K Peters, 2003.
Borwein, J. M.; Bradley, D. M.; Broadhurst,
D. J.; and Lisonek, P. "Special Values of Multidimensional Polylogarithms."
Trans. Amer. Math. Soc.
353
, 907-941, 2001.
Berndt, B. C.
Ramanujan's
Notebooks, Part IV.
New York: Springer-Verlag, pp. 323-326, 1994.
Cohen,
H.; Lewin, L.; and Zagier, D. "A Sixteenth-Order Polylogarithm Ladder."
Exper. Math.
1
, 25-34, 1992.
http://www.expmath.org/expmath/volumes/1/1.html
.
Erdélyi,
A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G.
Higher
Transcendental Functions, Vol. 1.
New York: Krieger, pp. 30-31,
1981.
Jonquière, A. "Ueber eine Klasse von Transcendenten,
welche durch mehrmahlige Integration rationaler Funktionen enstehen."
Öfversigt
af Kongl. Vetenskaps-Akademiens Förhandlingar
45
, 522-531, 1888.
Jonquière,
A. "Note sur la série
."
Öfversigt af Kongl.
Vetenskaps-Akademiens Förhandlingar
46
, 257-268, 1888.
Jonquière,
A. "Ueber einige Transcendente, welche bei den wiederholten Integration rationaler
Funktionen auftreten."
Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar
15
,
1-50, 1889.
Jonquière, A. "Note sur la série
."
Bull. Soc. Math. France
17
, 142-152, 1889.
Lewin, L.
Dilogarithms
and Associated Functions.
London: Macdonald, 1958.
Lewin, L.
Polylogarithms
and Associated Functions.
New York: North-Holland, 1981.
Lewin,
L. (Ed.).
Structural
Properties of Polylogarithms.
Providence, RI: Amer. Math. Soc., 1991.
Nielsen,
N.
Der Euler'sche Dilogarithms.
Leipzig, Germany: Halle, 1909.
Prudnikov,
A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized
Zeta Function
,
Bernoulli Polynomials
,
Euler Polynomials
,
and Polylogarithms
."
§1.2 in
Integrals
and Series, Vol. 3: More Special Functions.
Newark, NJ: Gordon and Breach,
pp. 23-24, 1990.
Sloane, N. J. A. Sequence
A073011
in "The On-Line Encyclopedia of Integer Sequences."
Truesdell,
C. "On a Function Which Occurs in the Theory of the Structure of Polymers."
Ann. Math.
46
, 114-157, 1945.
Zagier, D. "Special
Values and Functional Equations of Polylogarithms." Appendix A in
Structural
Properties of Polylogarithms
(Ed. L. Lewin). Providence, RI: Amer. Math.
Soc., 1991.
is used for the
and equal to
where
is the
and
are called the
for fixed complex
is not completely well-defined, since it depends on how
is approached in four-dimensional
-space.
The polylogarithm of
appears in the third-order correction
term in the gyromagnetic ratio of the electron.
The derivative of a polylogarithm is itself a polylogarithm,
A number of remarkable identities exist for polylogarithms, including the amazing identity satisfied by
,
where
(OEIS
."
Öfversigt af Kongl.
Vetenskaps-Akademiens Förhandlingar
46
, 257-268, 1888.
Jonquière,
A. "Ueber einige Transcendente, welche bei den wiederholten Integration rationaler
Funktionen auftreten."
Bihang till Kongl. Svenska Vetenskaps-Akademiens Handlingar
15
,
1-50, 1889.
Jonquière, A. "Note sur la série
."
Bull. Soc. Math. France
17
, 142-152, 1889.
Lewin, L.
,
Bernoulli Polynomials
,
Euler Polynomials
,
and Polylogarithms
."
§1.2 in