APPROXIMATION OF THE DILOGARITHM
FUNCTION
Dilogarithm Function
MEHDI HASSANI
Institute for Advanced Studies in Basic Sciences
P.O. Box 45195-1159, Zanjan, Iran.
EMail: mmhassany@srttu.edu
Mehdi Hassani
vol. 8, iss. 1, art. 25, 2007
Title Page
Received:
15 April, 2006
Accepted:
03 January, 2007
Communicated by:
A. Lupaş
2000 AMS Sub. Class.:
33E20.
Key words:
Special function, Dilogarithm function, Digamma function, Polygamma function, Polylogarithm function, Lerch zeta function.
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Abstract:
In this short
R xnote,t we approximate Dilogarithm function, defined by
dilog(x) = 1 log
1−t dt. Letting
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
1
D(x, N ) < dilog(x) < D(x, N ) + N
x
Dilogarithm Function
Mehdi Hassani
vol. 8, iss. 1, art. 25, 2007
hold true for all N ∈ N.
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Dedication:
Dedicated to Professor Yousef Sobouti on the occasion of his 75th birthday.
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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
Dilogarithm Function
Mehdi Hassani
vol. 8, iss. 1, art. 25, 2007
N
1
π2 X
D(x, N ) = − log2 x −
+
2
6
n=1
1
n2
+
1
log x
n
.
xn
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Aim of Present Work. The aim of this note is to prove that:
1
xN
Lower Bound. For every x > 0 and N ∈ N, let:
0 < dilog(x) − D(x, N ) <
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(x > 1, N ∈ N).
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L(x, N ) = dilog(x) − D(x, N ).
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A simple computation, yields that:
d
L(x, N ) = log x
dx
JJ
N +1
X 1
x
+
1 − x n=0 xn
∞
!
< log x
X 1
x
+
1 − x n=0 xn
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!
= 0.
So, L(x, N ) is a strictly decreasing
function of the variable x, for every N ∈ N. Con
sidering 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→+∞
Close
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
π2 X 1
π2
U(1, N ) =
−
−
1
=
Ψ(1,
N
+
1)
−
1
≤
− 2 < 0,
6
n2
6
n=1
in which Ψ(m, x) is the m-th polygamma function,
m-th derivative of the digamma
R ∞ the
d
−t x−1
function, Ψ(x) = dx log Γ(x), with Γ(x) = 0 e t dt (see [1, 2]). A simple
computation, yields that:
!
N
+1
X
d
N
x
1
U(x, N ) = log x
+
+ N +1 .
n
dx
1 − x n=0 x
x
To determine the sign of
d
U(x, N ),
dx
1. Suppose x > 1. Since,
log x
x−1
we distinguish two cases:
is strictly decreasing, we have
log x
log x
N ≥ 1 = lim
>
,
x→1 x − 1
x−1
P∞
N
1
N
1
which is log
>
or
equivalently
>
N
+1
n=N +2 xn , and this yields
x
x−1
x
log x
d
that dx
U(x, N ) > 0. So, U(x, N ) is strictly increasing for every N ∈ N. Thus,
U(x, N ) < 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.
Dilogarithm Function
Mehdi Hassani
vol. 8, iss. 1, art. 25, 2007
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x
x
2. Suppose 0 < x < 1 and N − log
≥ 0. We observe that 1 < log
< +∞ and
x−1
x−1
PN +1 1
N
+2
1−x
d
n=0 xn = xN +1 (1−x) . Considering these facts, we see that dx U(x, N ) and
x
N − log
have same sign; i.e.
x−1
d
log x
sgn
U(x, N ) = sgn N −
.
dx
x−1
Dilogarithm Function
Thus, U(x, N ) is increasing and so,
Mehdi Hassani
π2
U(x, N ) ≤ lim− U(x, N ) = Ψ(1, N + 1) − 1 ≤
− 2 < 0.
x→1
6
vol. 8, iss. 1, art. 25, 2007
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Connection with Other Functions. Using Maple, we have:
π2
1
log x
x−1
1
2
D(x, N ) = − log x −
+ 2 N +
− log
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
polylog(a, z) =
∞
X
zn
n=1
n
,
a
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,
Φ(z, a, v) =
∞
X
n=1
zn
,
(v + n)a
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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]).
Dilogarithm Function
Mehdi Hassani
vol. 8, iss. 1, art. 25, 2007
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References
[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 York-Amsterdam, 1981.
Dilogarithm Function
Mehdi Hassani
vol. 8, iss. 1, art. 25, 2007
Title Page
[5] E.W. WEISSTEIN, "Dilogarithm." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Dilogarithm.html
[6] E.W.
WEISSTEIN,
"Lerch
Wolfram
Web
Resource.
LerchTranscendent.html
Transcendent."
From
MathWorld–A
http://mathworld.wolfram.com/
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