EVALUATING A FAMILY OF INTEGRALS

Russell Euler

In the study of blackbody radiation, a relationship between the Stefan– Boltzmann constant and Planck’s constant can be derived using the fact that

(1)  ∫_{0}^{∞} \frac{x^{3}dx}{e^{x} − 1} = \frac{π^{4}}{15} .

The purpose of this paper is to obtain a generalization of identity (1). Consider the family of improper integrals defined by

I(p) = ∫_{0}^{∞} \frac{x^{p}dx}{e^{x} − 1}.

Conditions will be imposed on p to ensure the convergence of the integral. If p ≥ 1, then f(x) = x^{p}/(e^{x} − 1) has a removable singularity at x = 0. Now,

\frac{1}{e^{x} − 1} = \frac{1}{e^{x}(1 − e^{−x})} = e^{−x} ∑_{n=0}^{∞}(e^{−x})^{n}

for e^{−x} < 1. So, for x > 0,

\frac{1}{e^{x} − 1} = ∑_{n=0}^{∞} e^{−(n+1)x}

with the convergence being uniform on compact subsets of the interval of convergence. Hence, if p > 0, then

\frac{x^{p}}{e^{x} − 1} = ∑_{n=0}^{∞} x^{p}e^{−(n+1)x}

for x ≥ 0. From the definition of the Gamma function [4], it is known that

∫_{0}^{∞} x^{p}e^{−(n+1)x}dx = \frac{Γ(p + 1)}{(n + 1)^{p+1}} .

Therefore, for p > 0,

I(p) = Γ(p + 1) ∑_{n=0}^{∞} \frac{1}{(n + 1)^{p+1}}

= Γ(p + 1) ∑_{n=1}^{∞} \frac{1}{n^{p+1}} . (2)

If p is an odd positive integer, then the series in (2) converges and has been evaluated in [1] using various expansion techniques beginning with the logarithmic derivative of the infinite product expansion of sin x. In this paper, the series in (2), with p an odd positive integer, will be evaluated in closed form using residue theory. To this end, consider

∑_{n=1}^{∞} \frac{1}{n^{p+1}} = \frac{1}{2} ∑_{n=−∞}^{∞}′ \frac{1}{n^{p+1}}

where the prime attached to the summation indicates that the term corresponding to n = 0 is to be omitted. It has been shown in [3] that if f(z) satisfies

|f(z)| ≤ \frac{M}{|z|^{k}}

on C_{N} for all nonnegative integers N where k > 1 and M are constants independent of N, and C_{N} is the square with vertices at (N + 1/2)(±1 ± i), then

∑_{n=−∞}^{∞} f(n)

is the negative of the sum of the residues of πf(z) cot πz at the poles of f(z). So, to evaluate the summation in (2), take

f(z) = \frac{1}{z^{p+1}}

and use ∑′.

It is known from [2] that

z cot z = ∑_{n=0}^{∞} \frac{(−1)^{n}2^{2n}B_{2n}z^{2n}}{(2n)!}

for |z| < π, where the B_{2n}’s are the Bernoulli numbers of even index. Hence,

\frac{π cot πz}{z^{p+1}} = ∑_{n=0}^{∞} \frac{(−1)^{n}(2π)^{2n}B_{2n}z^{2n−p−2}}{(2n)!}

for 0 < |z| < 1. So,

Res_{z=0} \frac{π cot πz}{z^{p+1}} = \frac{(−1)^{(p+1)/2}(2π)^{p+1}B_{p+1}}{(p + 1)!} .

Therefore,

∑_{n=1}^{∞} \frac{1}{n^{p+1}} = −\frac{(−1)^{(p+1)/2}(2π)^{p+1}B_{p+1}}{2(p + 1)!} .

Substituting the latter identity into (2) and simplifying yields

I(p) = \frac{(−1)^{(p+3)/2}(2π)^{p+1}B_{p+1}}{2(p + 1)}

= \frac{(−1)^{(p+3)/2}2^{p}π^{p+1}B_{p+1}}{p + 1} . (3)

In conclusion, identity (3) is valid for all odd positive integers p and provides a generalization of (1). In particular, for p = 3, equation (3) becomes

I(3) = −2π^{4}B_{4} = −2π^{4}(−1/30) = π^{4}/15,

which agrees with (1).

\underline{References}

1. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer–Verlag, 1982.

2. Y. Luke, The Special Functions and Their Approximations, Vol. 1, Academic Press, 1969.

3. M. Spiegel, Complex Variables with an Introduction to Conformal Mapping and Its Applications, Schaum’s Outline Series, McGraw–Hill, 1964.

4. M. Spiegel, Theory and Problems of Laplace Transforms, Schaum’s Outline Series, McGraw–Hill, 1965.

Russell Euler Department of Mathematics and Statistics Northwest Missouri State University Maryville, MO 64468-6001 email: 0100120@acad.nwmissouri.edu

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