What do Taylor coefficients tell us?

The Taylor series f(z):= \Sum _0^oo a_n z^n with, say, the radius of convergence 1, encodes all the information about f in the sequence {a_n}. The problem is how are we going to recover it? How one can tell , e.g., that f is a rational function of z (L. Kronecker, 1881) ( of course, f being a polynomial is seen at once). Or , why a_n= cos (sqrt {n}) , for all n, tells us that f is analytic in the whole complex plane except z=1 (l. Leau, 1899), while a_n =cos (n), when n =k^2, k being an integer, and 0 otherwise, tells us that f (z) will go nowhere beyond the unit circle (E. Fabry, 1896). More recently, if a_n=g(n), g is a bounded analytic function in the sector |arg z| (less than) \phi \leq \pi/2, then f extends to the heart-shaped region (2\pi-cot\phi) log |z| (greater than) arg z (greater than) cot ( \phi) log |z| (yours truly, 1997). These are the questions and answers that make, ok, arguably, complex analysis the most beautiful part of mathematics. The talk should be accessible to everyone who has ever had a one semester of complex variables and it is directed at students . -Dr. Dmitry Khavinson