Stirling-Ramanujan constants are exponential periods

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 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 is 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 ℚ(t,e^-t).