[email protected] Received 15 April, 2006

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Volume 8 (2007), Issue 1, Article 25, 3 pp.

APPROXIMATION OF THE DILOGARITHM FUNCTION

MEHDI HASSANI

INSTITUTE FORADVANCEDSTUDIES INBASICSCIENCES

P.O. BOX45195-1159 ZANJAN, IRAN. [email protected]

Received 15 April, 2006; accepted 03 January, 2007 Communicated by A. Lupa¸s

Dedicated to Professor Yousef Sobouti on the occasion of his 75th birthday.

ABSTRACT. In this short note, we approximate Dilogarithm function, defined bydilog(x) = Rx

1 logt

1−tdt. Letting

D(x, N) =−1

2log²x−π² 6 +

N

X

n=1 1

n²+_n¹logx

xⁿ ,

we show that for everyx >1, the inequalities

D(x, N)<dilog(x)<D(x, N) + 1 x^N hold true for allN ∈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 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

x^N⁺¹

,

where

D(x, N) =−1

2log²x− π² 6 +

N

X

n=1 1

n² + ¹_nlogx

xⁿ .

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2 MEHDIHASSANI

Aim of Present Work. The aim of this note is to prove that:

0<dilog(x)− D(x, N)< 1

x^N (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 every N ∈ N. Considering L(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.

Upper Bound. For everyx >0andN ∈N, let:

U(x, N) = dilog(x)− D(x, N)− 1 x^N. 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) = _dx^d log Γ(x), withΓ(x) =R∞

0 e^−tt^x−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

x^N⁺¹.

To determine the sign of _dx^dU(x, N), we distinguish two cases:

(1) Supposex >1. Since, ^log_x−1^x is strictly decreasing, we have

N ≥1 = lim

x→1

logx

x−1 > logx x−1, which is _log^N_x > _x−1¹ or equivalently _xN+1^Nlogx > P∞

n=N+2 1

xⁿ, and this yields that

d

dxU(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.

(2) Suppose 0 < x < 1 and N − ^log_x−1^x ≥ 0. We observe that 1 < ^log_x−1^x < +∞ and PN+1

n=0 1

xⁿ = _xN+1^1−x^N+2(1−x). Considering these facts, we see that _dx^dU(x, N) andN − ^log_x−1^x have same sign; i.e.

sgn d

dxU(x, N)

= sgn

N − logx x−1

.

J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 25, 3 pp. http://jipam.vu.edu.au/

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APPROXIMATION OF THEDILOGARITHMFUNCTION 3

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

N²x^N + logx N x^N −log

x−1 x

logx

+ polylog

2,1 x

−logx x^N Φ

1 x,1, N

− 1 x^NΦ

1 x,2, N

,

in which

polylog(a, z) =

∞

X

n=1

zⁿ n^a,

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,

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]).

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.

[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/