  Ramanujan studied a general class of Stirling constants that are the
resummation of some natural divergent series. These constants include the
classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find
natural integral representations for all these constants that appear as
exponential periods in the field $\mathbb Q (t,e^{-t})$ which reveals their
natural transalgebraic nature. We conjecture that all these constants are
transcendental numbers. Euler-Mascheroni's and Stirling's integral formula are
classical, but the integral formula for Glaisher-Kinkelin appears to be new, as
well as the integral formulas for the higher Stirling-Ramanujan constants. The
method presented generalizes naturally to prove that many other constants are
exponential periods over the field $\mathbb Q(t,e^{-t})$.
