EVALUATION OF A CLASS OF MULTIPLE INTEGRALS

Mohamed Akkouchi

Abstract. In his paper [1], K. L. D. Gunawardena introduced two sequences of multiple integrals (see (1.1) and (1.2) below) encountered in one of his courses of mathematical statistics. Using a key probability result, he evaluated these integrals. At the end of [1], he asked the question whether these integrals could be computed without using results from probability theory. The aim of this paper is to answer that question and make direct computations of these integrals by elementary methods without using probabilistic tools.

1. Introduction. In his paper [1], K. L. D. Gunawardena has considered the following sequences of multiple integrals

I_{n} := ∫_{0}^{1} ⋯ ∫_{0}^{1} \frac{θ^{n}(x_{1} ⋯ x_{n})^{θ−1}}{ln(x_{1} ⋯ x_{n})} dx_{1} ⋯ dx_{n}  (1.1)

and

J_{n} := ∫_{0}^{∞} ⋯ ∫_{0}^{∞} \frac{θ^{n}[(1 + x_{1}) ⋯ (1 + x_{n})]^{−(θ+1)}}{ln[(1 + x_{1}) ⋯ (1 + x_{n})]} dx_{1} ⋯ dx_{n}  (1.2)

where θ > 0 and n is an integer greater than or equal to 2. He computed separately, (1.1) and (1.2), by using a key probability result and asked whether (1.1) and (1.2) could be computed without making use of results from probability theory.

The purpose of this note is to make direct computations of (1.1) and (1.2) by using tools only from classical analysis. More precisely, we compute (1.1) and (1.2) by using some convenient and natural changes of variables. Before starting computations, we need to introduce the following sequence of multiple integrals:

K_{n} := ∫_{0}^{∞} ⋯ ∫_{0}^{∞} \frac{θ^{n} exp (−θ(y_{1} + ⋯ + y_{n}))}{y_{1} + ⋯ + y_{n}} dy_{1} ⋯ dy_{n},  (1.3)

for all integers n greater than or equal to 2. The introduction of these integrals is justified by the following observation.

\underline{Observation}. For all integers n greater than or equal to 2, we have the following equalities:

−I_{n} = J_{n} = K_{n}.  (1.4)

To prove the first equality in (1.4), it is sufficient to use the following change of variable:

y_{i} := \frac{1}{1 + x_{i}} , for all i = 1, 2, ..., n.  (1.5)

To prove the second equality in (1.4), it is sufficient to use the following change of variable:

y_{i} := ln(1 + x_{i}), for all i = 1, 2, ..., n.  (1.6)

By this observation, we are led to compute only the integrals K_{n} for n ≥ 2. It is clear that K_{1} is infinite. To compute K_{n}, we shall use the spherical coordinates in the Euclidean space ℝ^{n}.

2. Recalls on Spherical Coordinates. In the Euclidean space ℝ^{n}, the spherical coordinates (r, ϕ_{1}, …, ϕ_{n−1}) correponding to the point (x_{1}, …, x_{n}) are given by

x_{1} = r sin ϕ_{n−1} ⋯ sin ϕ_{3} sin ϕ_{2} sin ϕ_{1}
x_{2} = r sin ϕ_{n−1} ⋯ sin ϕ_{3} sin ϕ_{2} cos ϕ_{1}
x_{3} = r sin ϕ_{n−1} ⋯ sin ϕ_{3} cos ϕ_{2}  (2.1)
⋯ = · · · · · · · · · · · ·
x_{n−1} = r sin ϕ_{n−1} cos ϕ_{n−2}
x_{n} = r cos ϕ_{n−1},

where 0 ≤ r < ∞; 0 ≤ ϕ_{1} < 2π; and 0 ≤ ϕ_{k} < π for all k = 2, … n − 1. Conversely, for all k = 1, 2, …, n − 1, by setting r_{k}^{2} := x_{1}^{2} + ⋯ + x_{k}^{2} and r_{n} := r, we have

cos(ϕ_{k}) = \frac{x_{k+1}}{r_{k+1}} ,  (2.2)

and

sin(ϕ_{k}) = \frac{r_{k}}{r_{k+1}} .  (2.3)

We recall that the Jacobian of this change of variables is given by

r^{n−1} sin^{n−2} ϕ_{n−1} sin^{n−3} ϕ_{n−2} ⋯ sin ϕ_{2}.  (2.4)

Let 𝕊_{n} be the unit sphere (i.e., the set of x = (x_{1}, ..., x_{n}) with ∥x∥ = 1, where ∥.∥ is the usual Euclidean norm of ℝ^{n}). Then (2.4) means that for any Lebesgue integrable function f on ℝ^{n}, we have

∫_{ℝ^{n}} f(x) dx = ∫_{0}^{∞} [∫_{𝕊_{n}} f(ru) dσ(u)] r^{n−1} dr,  (2.5)

where x = ru ∈ ℝ^{n} = [0, ∞) × 𝕊_{n} and

dσ(u) = sin^{n−2} ϕ_{n−1} sin^{n−3} ϕ_{n−2} ⋯ sin ϕ_{2} dϕ_{1} ⋯ dϕ_{n−2} dϕ_{n−1}  (2.6)

is the restriction of the Lebesgue measure on 𝕊_{n}.

Let us consider the set ℝ_{+}^{n} := {x = (x_{1}, …, x_{n}) ∈ ℝ^{n} : 0 ≤ x_{i} ≤ ∞, for all i = 1, …, n}, and denote

𝕊_{+}^{n} := \{u = (1, ϕ_{1}, ⋯, ϕ_{n−1}) ∈ 𝕊_{n} : 0 ≤ ϕ_{i} ≤ \frac{π}{2} , for all i = 1, …, n − 1\}.

Then, it is clear that for all x = ru ∈ ℝ^{n}, we have

x ∈ ℝ_{+}^{n} ⟺ u ∈ 𝕊_{+}^{n},  (2.7)

and for every Lebesgue integrable function f on ℝ_{+}^{n}, we have

∫_{ℝ_{+}^{n}} f(x) dx = ∫_{0}^{∞} [∫_{𝕊_{+}^{n}} f(ru) dσ(u)] r^{n−1} dr.  (2.8)

3. Computation of K_{n}, I_{n}, and J_{n}. First, we start by setting y_{i} := x_{i}^{2} where 0 ≤ x_{i} < ∞ for all i = 1, 2, …, n. With these variables, we have

dy_{1} ⋯ dy_{n} = 2^{n}x_{1} ⋯ x_{n}dx_{1} ⋯ dx_{n},  (3.1)

and then

K_{n} = 2^{n}θ^{n} ∫_{0}^{∞} ⋯ ∫_{0}^{∞} \frac{exp (−θ(x_{1}^{2} + ⋯ + x_{n}^{2}))}{x_{1}^{2} + ⋯ + x_{n}^{2}} x_{1} ⋯ x_{n}dx_{1} ⋯ dx_{n}. (3.2)

Now, we use the spherical coordinates, and set as in (2.1)

x_{1} = r sin ϕ_{n−1} ⋯ sin ϕ_{3} sin ϕ_{2} sin ϕ_{1}
x_{2} = r sin ϕ_{n−1} ⋯ sin ϕ_{3} sin ϕ_{2} cos ϕ_{1}
x_{3} = r sin ϕ_{n−1} ⋯ sin ϕ_{3} cos ϕ_{2}  (3.3)
⋯ = · · · · · · · · · · · ·
x_{n−1} = r sin ϕ_{n−1} cos ϕ_{n−2}
x_{n} = r cos ϕ_{n−1},

where 0 ≤ r < ∞ and 0 ≤ ϕ_{i} ≤ \frac{π}{2} , for all i = 1, …, n − 1. Then, from (3.2), (3.3), and Fubini’s Theorem, we get

K_{n} = 2^{n}θ^{n} [∫_{0}^{∞} r^{2n−3} exp (−θr^{2}) dr] ∏_{i=1}^{n−1} [∫0^{\frac{π}{2}} sin^{2i−1}(ϕ_{i}) cos(ϕ_{i}) dϕ_{i}] . (3.4)

The first integral in (3.4) is easily computed by setting t := θr^{2}. Indeed, with this change of variable, we have:

∫_{0}^{∞} r^{2n−3} exp (−θr^{2}) dr = \frac{1}{2θ^{n−1}} ∫_{0}^{∞} t^{n−2} exp (−t) dt = \frac{Γ(n − 1)}{2θ^{n−1}} = \frac{(n − 2)!}{2θ^{n−1}}.

(3.5)

The product in (3.4) is easy to obtain. Indeed, we have

∏_{i=1}^{n−1} [∫0^{\frac{π}{2}} sin^{2i−1}(ϕ_{i}) cos(ϕ_{i}) dϕ_{i}] = ∏_{i=1}^{n−1} \frac{1}{2i} = \frac{1}{2^{n−1}(n − 1)!}.  (3.6)

From (3.4), (3.5), and (3.6), we conclude that

K_{n} = \frac{θ}{n − 1}.  (3.7)

We deduce that

I_{n} = −\frac{θ}{n − 1},  (3.8)

and

J_{n} = \frac{θ}{n − 1}.  (3.9)

So we recapture (in (3.8) and (3.9)) the results already obtained by K. L. D. Gunawardena in [1] by using probabilistic tools.

\underline{Acknowledgment}. The author wishes to thank the referees for their helpful comments.

\underline{References}

1. K. L. D. Gunawardena, “On the Evaluation of Multiple Integrals,” Missouri Journal of Mathematical Sciences, 6 (1994), 29–33.

Mathematics Subject Classification (2000) 26B10, 26B15, 26B20, 28A35

Mohamed Akkouchi Department of Mathématiques Faculté des Sciences-Semlalia Université Cadi Ayyad Marrakech. Maroc. (MOROCCO) email: akkouchimo@yahoo.fr