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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.

**Occasions:** Fields institute, Toronto, February 2009

**Documents:** slideshow, TeXmacs
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