May 5, 1993 - GRADSHTEYN, I.S. and RYZHIK, I.M. Tables of Integrals, Series, and Products (Academic. Press, 1980). 6. GUMBEL, E.J. A quick estimation of ...
405
Internat. J. Math. & Math. Sci. VOL. 17 NO. 2 (1994) 405-408
CERTAIN INTEGRALS INVOLVING LOGARITHMIC AND EXPONENTIAL FUNCTIONS M. ASLAM CHAUDHRY and M. AHMAD
Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia (Received September 20, 1991 and in revised form May 5, 1993)
ABSTRACT. In this paper
we have
evaluated the integrals
x’- In
x
exp(-ax
bx- )dx
and
jfo for all n
x’-:(ax
b)(ln x) exp(-ax
1,2,3,.... Some applications of the results
bx-’)dx
are discussed
and an open problem is posed.
KEY WORDS AND PHRASES. Macdonald functions, logarithmic and exponential function, statistical inference theory. 1991 AMS SUBJECT CLASSIFICATION: 33-A-10. 1.
INTRODUCTION. The integrals x ’-a
In
x
exp(-ax
bx -)dx,
a>O,b. >0,
(1)
and
fo
x"-:(ax
b)(ln x) exp(-ax bx-’)dx, a > O, b > O, (n 1,2,3,....)
(2)
arise in statistical inference theory when Frechet and other distributions are applied to solve impor-
tant problems related to ocean engineering technology, resource management, and weather phenom-
forecasting of wind velocity, flood, rainfall, etc. See [3] and [6,7,8]. The closed form solutions of the integrals (1) and (2) are not known for any n 1,2,3,.... Some numerical techniques are used to evaluate these integrals [2,8]. We have evaluated these integrals (1) and (2). Some special cases of the results are also discussed. ena such as estimation and
M.A. CHAUDHRY AND M. AHMAD
406
LEMMA. For n 0.1,2,3
2.
Leil)nitz rule of differentiation and from the recursion form,las
PROOF. This follows from the
[5, p 970]. 3. TItEOREM. ’:x
’- In
X
exp(--ax-bx-1)dx
a"
j(n
3--1
(a > 0, b > 0,
(4)
).
=0,1,2,3
n
PROOF. It is known that [5, p 31a]
j/oX
-1
2(b/a)O/Ko(2-)
bx-1)dx
exp(--ax
(5)
a>0, b>0. Performing formal differentiation with respect to the parameter c we obtain the equality
/aOx-’ In x
x
exp(-ax- bx-’)dx
2(b/a) ’/:
-ln(b/a)K(2v/) + K(2)
The process of the formal differentiation is justified
[4, pp. 427-448].
Using the integral representation [5, p. 358] for the Macdonald function K, 0
e- ch(t)sinh(at)dt
Oa [K,(z)]
(6)
we
get
(7)
which implies
0
y(z)]o-o
o
[ -lln x exp(-ax- bx-1)dx
ln
Oa Letting a
0 in
(6) and using (8)
we
(s)
get
JO
b/ a Ko( 2 V/) (a > O, b > O).
We
can rewrite
(9)
(9)
in the operational form as follows:
L
{ lnX exp(-bx-1); a}
where L is the Laplace transform operator
[1].
ln(b/a)I(2v/-)’
(o)
407
INTEGRALS INVOLVING SPECIAL FUNCTIONS
By using the lemma and property [1]
(-ad), f(a)
L{x’f(z);a}
of the Laplace transformation we get the result from
(11)
(n= 1,2,3,...) (9) and (11).
COROLLARY 1. See [5, p. 577]
foexp(-t(x/c + c/xlllnXdxx
2(ln c) Ko(2/t), (c > 0,
PROOF. This follows from (9)
when we take a
> 0).
i/c and b =/c.
COROLLARY 2.
[
exp(3t
2a cosh 3t)dt
1/a32Ko(2a) ( >0, >0).
PROOF. This follows from (4) when we take n 1, b cm, > 0. COROLLARY 3. For all n 0, 1,2,3,... and a > 0, b > 0,
x"-2(ax a=
b)(ln x) exp(-ax
a
and use the transformation x
bx-1)dx
)] (ba)/21n(b/a)K(2) +n (ab)"-’)/2lQj(n j) ]
,=,
(12)
PROOF. This follows by applying integration by parts to the integral in (4). In particular for n 0, a=g/c,b=gc, g>0, c > 0 in (12), we get
(ln x) ( > 0, c> 0). We state here an open problem. The solution to the problem will he far-reaching consequences in statistical inference theory. It should be noted that the solution to the problem is not known even for n
2 and for any value of
4. STATEMENT
.
OF THE OPEN PROBLEM.
Evaluate
(/n x)’x ’-’ exp(-ax
ACKNOWLEDGEMENTS. The authors
bx-’)dx
are indebted to the
(a > 0, b > 0,
n
> 2).
KFUPM for the excellent research
facilities. The helpful comments made by the referee contributed to the improvement of the paper
and are appreciated.
M.A. CHAUDHRY AND M. AHMAD
408
REFERENCES
of Integral Transforms, vol. I, (McGraw-ltill, 1954).
1.
ERDELYI, A., et al., Tables
2.
CIIAUDIIRY, M.A. and AIIMAD, M. On
3.
FISHER, R.A. and TIPPETT, L.tt. Limiting forms of the frequency distributions of the largest
a probability function useful in size modeling, Accepted for publication in Canadian Journal of Forest Research (1992).
or smallest member of a
sample, Proceedzngs
of the
Cambridge Philosophical Society,
24(1928), 180-190.
BUDAK, B.M. and FOMIN, S.V. Multiple Integrals, Field Theory and Series, (Mir Publishers, 1978). 5. GRADSHTEYN, I.S. and RYZHIK, I.M. Tables of Integrals, Series, and Products (Academic Press, 1980).
4.
6.
GUMBEL, E.J. A quick estimation of the parameters of Frechet distribution, Review of the International Statistical Institute, 33(3), (1965), 349-363.
7.
SINGH, N.P. and SINGH, K.P.
8.
SINGH, N.P.
Estimation of Frechet distribution parameters by joint distribution of m’ extremes, Statistica, Anno, L., 1(1990), 59-68. Maximum likelihood estimation of Frechet distribution parameters, Journal Statistical Studies, 7(1987), 11-22.
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