There are complex functions with the same natural boundaries in the complex plane, but,they are different from each other. For example, there are lots of different lacunary power series with different integral coefficients,different power of nomials but they have the same natural boundary.


As we know, some functions in complex plane can be distinguished or characterized by their poles.


Now,the question is:how to characterize or distinguish or classify those function with same natural boundary from each other by information of their behavior going to the boundary,that is asymptotic behaviour,or something like poles of function. Can they be distinguished or characterized only by asymptotic behavior. Any reference?


A question like this is posted on Mathematics,but possibly get no answer,so I ask it here.



How to classify the complex function with same natural boundary in complex plane?
There is a paper of Breuer and Simon, "Natural Boundaries and Spectral Theory" (some slides [here](http://www.mth.kcl.ac.uk/~pushn/ebd65/simon.pdf) ). They give, among other things, the definition of "strong natural boundary". This concept relates to "right limits" of the sequence of Taylor coefficients and give a criterion based on parts of the integrability of the original function on the boundary.


I should also mention that these techniques can lead to a notion of "analytic continuation" through the natural frontier for a natural class of series (Sauzin and Tiozzo, ["Generalised continuation by means of right limits"](http://arxiv.org/abs/1301.1175) ).


