Volume 8 (2007), Issue 1, Article 25, 3 pp.
APPROXIMATION OF THE DILOGARITHM FUNCTION
MEHDI HASSANI
I NSTITUTE FOR A DVANCED S TUDIES IN BASIC S CIENCES
P.O. B OX 45195-1159
Z ANJAN , I RAN .
mmhassany@srttu.edu
Received 15 April, 2006; accepted 03 January, 2007
Communicated by A. Lupaş
Dedicated to Professor Yousef Sobouti on the occasion of his 75th birthday.
. In this short note, we approximate Dilogarithm function, defined by dilog(x) =
RAxBSTRACT
log t
dt.
Letting
1 1−t
N
1
π2 X
D(x, N ) = − log2 x −
+
2
6
n=1
1
n2
+ n1 log x
,
xn
we show that for every x > 1, the inequalities
D(x, N ) < dilog(x) < D(x, N ) +
1
xN
hold true for all N ∈ N.
Key words and phrases: Special function, Dilogarithm function, Digamma function, Polygamma function, Polylogarithm
function, Lerch zeta function.
2000 Mathematics Subject Classification. 33E20.
Definition. The Dilogarithm function dilog(x) is defined for every x > 0 as follows [5]:
Z x
log t
dilog(x) =
dt.
1 1−t
Expansion. The following expansion holds true when x tends to infinity:
1
dilog(x) = D(x, N ) + O
,
xN +1
where
N
1
π 2 X n12 + n1 log x
D(x, N ) = − log2 x −
+
.
n
2
6
x
n=1
113-06
2
M EHDI H ASSANI
Aim of Present Work. The aim of this note is to prove that:
1
0 < dilog(x) − D(x, N ) < N
(x > 1, N ∈ N).
x
Lower Bound. For every x > 0 and N ∈ N, let:
L(x, N ) = dilog(x) − D(x, N ).
A simple computation, yields that:
d
L(x, N ) = log x
dx
N +1
X 1
x
+
1 − x n=0 xn
!
∞
< log x
X 1
x
+
1 − x n=0 xn
!
= 0.
So, L(x, N ) is a strictly decreasing function of the variable x, for every N ∈ N. Considering
L(x, N ) = O xN1+1 , we obtain a desired lower bound for the Dilogarithm function, as follows:
L(x, N ) > lim L(x, N ) = 0.
x→+∞
Upper Bound. For every x > 0 and N ∈ N, let:
U(x, N ) = dilog(x) − D(x, N ) −
1
.
xN
First, we observe that
N
U(1, N ) =
π2 X 1
π2
−
−
1
=
Ψ(1,
N
+
1)
−
1
≤
− 2 < 0,
2
6
n
6
n=1
in which Ψ(m, x) is the m-th polygamma function, the m-th derivative of the digamma function,
R∞
d
Ψ(x) = dx
log Γ(x), with Γ(x) = 0 e−t tx−1 dt (see [1, 2]). A simple computation, yields that:
!
N
+1
X
x
1
N
d
U(x, N ) = log x
+
+ N +1 .
n
dx
1 − x n=0 x
x
d
U(x, N ), we distinguish two cases:
dx
x
Since, log
is strictly decreasing, we have
x−1
To determine the sign of
(1) Suppose x > 1.
log x
log x
>
,
x→1 x − 1
x−1
P∞
1
1
N
which is log
> x−1
or equivalently xN +1Nlog x >
n=N +2 xn , and this yields that
x
d
U(x, N ) > 0. So, U(x, N ) is strictly increasing for every N ∈ N. Thus, U(x, N ) <
dx
N ≥ 1 = lim
limx→+∞ U(x, N ) = 0; as desired in this case. Also, note that in this case we obtain
U(x, N ) > U(1, N ) = Ψ(1, N + 1) − 1.
x
x
(2) Suppose 0 < x < 1 and N − log
≥ 0. We observe that 1 < log
< +∞ and
x−1
x−1
PN +1 1
log x
1−xN +2
d
n=0 xn = xN +1 (1−x) . Considering these facts, we see that dx U(x, N ) and N − x−1
have same sign; i.e.
d
log x
sgn
U(x, N ) = sgn N −
.
dx
x−1
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 25, 3 pp.
http://jipam.vu.edu.au/
A PPROXIMATION OF THE D ILOGARITHM F UNCTION
3
Thus, U(x, N ) is increasing and so,
U(x, N ) ≤ lim− U(x, N ) = Ψ(1, N + 1) − 1 ≤
x→1
π2
− 2 < 0.
6
Connection with Other Functions. Using Maple, we have:
π2
1
log x
x−1
1
2
+ 2 N +
− log
log x
D(x, N ) = − log x −
2
6
N x
N xN
x
1
log x
1
1
1
+ polylog 2,
− N Φ
, 1, N − N Φ
, 2, N ,
x
x
x
x
x
in which
∞
X
zn
polylog(a, z) =
,
a
n
n=1
is the polylogarithm function of index a at the point z and defined by the above series if |z| < 1,
and by analytic continuation otherwise [4]. Also,
∞
X
zn
Φ(z, a, v) =
,
(v + n)a
n=1
is the Lerch zeta (or Lerch-Φ) function defined by the above series for |z| < 1, with v 6=
0, −1, −2, . . . , and by analytic continuation, it is extended to the whole complex z-plane for
each value of a and v (see [3, 6]).
R EFERENCES
[1] M. ABRAMOWITZ AND I.A. STEGUN, Handbook of Mathematical Functions: with Formulas,
Graphs, and Mathematical Tables, Dover Publications, 1972.
[2] N.N. LEBEDEV, Special Functions and their Applications, Translated and edited by Richard A.
Silverman, Dover Publications, New York, 1972.
[3] L. LEWIN, Dilogarithms and associated functions, MacDonald, London, 1958.
[4] L. LEWIN, Polylogarithms and associated functions, North-Holland Publishing Co., New YorkAmsterdam, 1981.
[5] E.W. WEISSTEIN, "Dilogarithm." From MathWorld–A Wolfram Web Resource. http://
mathworld.wolfram.com/Dilogarithm.html
[6] E.W. WEISSTEIN, "Lerch Transcendent." From MathWorld–A Wolfram Web Resource. http:
//mathworld.wolfram.com/LerchTranscendent.html
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 25, 3 pp.
http://jipam.vu.edu.au/