  One solution to a relatively recent American Mathematical Monthly problem
[6], requesting the evaluation of a real definite integral, could be couched in
terms of a contour integral which vanishes {\textit{a priori.}} While the
required real integral emerged on setting to zero the real part of the contour
quadrature, the obligatory, simultaneous vanishing of the imaginary part
alluded to still another pair of real integrals forming the first two entries
in the infinite log-sine sequence, known in its entirety. It turns out that
identical reasoning, utilizing the same contour but a slightly different
analytic function thereon, sufficed not only to evaluate that sequence anew, on
the basis of a vanishing real part, but also, in setting to zero its conjugate
imaginary part, to recover the fundamental Bernoulli number recurrence. The
even order Bernoulli numbers $B_{2k}$ entering therein were revealed on the
basis of their celebrated connection to Riemann's zeta function $\zeta(2k).$
Conversely, by permitting the related Bernoulli polynomials to participate as
integrand factors, Euler's connection itself received an independent
demonstration, accompanied once more by an elegant log-sine evaluation,
alternative to that already given. And, while the Bernoulli recurrence is
intended to enjoy here the pride of place, this note ends on a gloss wherein
all the motivating real integrals are recovered yet again, and in quite
elementary terms, from the Fourier series into which the Taylor development for
Log$(1-z)$ blends when its argument $z$ is restricted to the unit circle.
