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Periods are defined as integrals of semialgebraic functions defined over therationals. Periods form a countable ring not much is known about. Examples aregiven by taking the antiderivative of a power series which is algebraic overthe polynomial ring over the rationals and evaluate it at a rational number. Wefollow this path and close these algebraic power series under taking iteratedantiderivatives and nearby algebraic and geometric operations. We obtain asystem of rings of power series whose coefficients form a countable real closedfield. Using techniques from o-minimality we are able to show that every periodbelongs to this field. In the setting of o-minimality we define exponentialintegrated algebraic numbers and show that the Euler constant is an exponentialintegrated algebraic number. Hence they are a good candiate for a naturalnumber system extending the period ring and containing important mathematicalconstants.

Periods, Power Series, and Integrated Algebraic Numbers