It is well known that Hardy fields can be extended with integrals, exponentials and solutions to Pfaffian first order differential equations . From the formal point of view, the theory of transseries allows for the resolution of more general algebraic differential equations. However, until now, this theory did not admit a satisfactory analytic counterpart. In this paper, we will introduce the notion of a transserial Hardy field. Such fields combine the advantages of Hardy fields and transseries. In particular, we will prove that the field of differentially algebraic transseries over carries a transserial Hardy field structure. Inversely, we will give a sufficient condition for the existence of a transserial Hardy field structure on a given Hardy field.

Keywords: transseries, Hardy field, o-minimality, algebraic differential equation

A.M.S. subject classification: 26A12, 34E05, 40A05, 03C64

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Revised version (2017, February, 7): Pdf

Acknowledgment: Lou van den Dries and Mickaël Matusinski kindly provided me with feedback that has given rise to the revised version with various corrections and improvements concerning the readability.