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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.

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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

2log2xπ2 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.

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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.

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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.

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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→1U(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,

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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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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 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

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