Dilogarithm Function Mehdi Hassani vol. 8, iss. 1, art. 25, 2007
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APPROXIMATION OF THE DILOGARITHM FUNCTION
MEHDI HASSANI
Institute for Advanced Studies in Basic Sciences P.O. Box 45195-1159, Zanjan, Iran.
EMail:mmhassany@srttu.edu
Received: 15 April, 2006
Accepted: 03 January, 2007
Communicated by: A. Lupa¸s 2000 AMS Sub. Class.: 33E20.
Key words: Special function, Dilogarithm function, Digamma function, Polygamma func- tion, Polylogarithm function, Lerch zeta function.
Dilogarithm Function Mehdi Hassani vol. 8, iss. 1, art. 25, 2007
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Close Abstract: In this short note, we approximate Dilogarithm function, defined by
dilog(x) =Rx 1
logt
1−tdt. Letting
D(x, N) =−1
2log2x−π2 6 +
N
X
n=1 1
n2 +1nlogx
xn ,
we show that for everyx >1, the inequalities
D(x, N)<dilog(x)<D(x, N) + 1 xN
hold true for allN ∈N.
Dedication: Dedicated to Professor Yousef Sobouti on the occasion of his 75th birthday.
Dilogarithm Function Mehdi Hassani vol. 8, iss. 1, art. 25, 2007
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Definition. The Dilogarithm functiondilog(x)is defined for everyx >0as follows [5]:
dilog(x) = Z x
1
logt 1−tdt.
Expansion. The following expansion holds true whenxtends to infinity:
dilog(x) =D(x, N) +O 1
xN+1
,
where
D(x, N) =−1
2log2x− π2 6 +
N
X
n=1 1
n2 + 1nlogx
xn .
Aim of Present Work. The aim of this note is to prove that:
0<dilog(x)− D(x, N)< 1
xN (x >1, N ∈N).
Lower Bound. For everyx >0andN ∈N, let:
L(x, N) = dilog(x)− D(x, N).
A simple computation, yields that:
d
dxL(x, N) = logx x 1−x+
N+1
X
n=0
1 xn
!
<logx x 1−x+
∞
X
n=0
1 xn
!
= 0.
So,L(x, N)is a strictly decreasing function of the variablex, for everyN ∈N. Con- sideringL(x, N) =O xN+11
, we obtain a desired lower bound for the Dilogarithm function, as follows:
L(x, N)> lim
x→+∞L(x, N) = 0.
Dilogarithm Function Mehdi Hassani vol. 8, iss. 1, art. 25, 2007
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Upper Bound. For everyx >0andN ∈N, let:
U(x, N) = dilog(x)− D(x, N)− 1 xN. First, we observe that
U(1, N) = π2 6 −
N
X
n=1
1
n2 −1 = Ψ(1, N+ 1)−1≤ π2
6 −2<0,
in whichΨ(m, x)is them-th polygamma function, them-th derivative of the digamma function, Ψ(x) = dxd log Γ(x), with Γ(x) = R∞
0 e−ttx−1dt (see [1, 2]). A simple computation, yields that:
d
dxU(x, N) = logx x 1−x +
N+1
X
n=0
1 xn
!
+ N
xN+1.
To determine the sign of dxdU(x, N), we distinguish two cases:
1. Supposex >1. Since, logx−1x is strictly decreasing, we have
N ≥1 = lim
x→1
logx
x−1 > logx x−1, which is logNx > x−11 or equivalently xN+1Nlogx > P∞
n=N+2 1
xn, and this yields that dxdU(x, N)>0. So,U(x, N)is strictly increasing for everyN ∈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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2. Suppose0< x < 1andN − logx−1x ≥0. We observe that1 < logx−1x <+∞and PN+1
n=0 1
xn = xN+11−x(1−x)N+2 . Considering these facts, we see that dxdU(x, N) and N − logx−1x have same sign; i.e.
sgn d
dxU(x, N)
= sgn
N − logx x−1
.
Thus,U(x, N)is increasing and so, U(x, N)≤ lim
x→1−U(x, N) = Ψ(1, N+ 1)−1≤ π2
6 −2<0.
Connection with Other Functions. Using Maple, we have:
D(x, N) =−1
2log2x− π2
6 + 1
N2xN + logx N xN −log
x−1 x
logx + polylog
2, 1
x
− logx xN Φ
1 x,1, N
− 1 xNΦ
1 x,2, N
,
in which
polylog(a, z) =
∞
X
n=1
zn na,
is the polylogarithm function of indexaat the pointzand 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,
Dilogarithm Function Mehdi Hassani vol. 8, iss. 1, art. 25, 2007
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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 ofaandv (see [3,6]).
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References
[1] M. ABRAMOWITZ AND I.A. STEGUN, Handbook of Mathematical Func- tions: 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.
[5] E.W. WEISSTEIN, "Dilogarithm." From MathWorld–A Wolfram Web Re- source.http://mathworld.wolfram.com/Dilogarithm.html [6] E.W. WEISSTEIN, "Lerch Transcendent." From MathWorld–A
Wolfram Web Resource. http://mathworld.wolfram.com/
LerchTranscendent.html