understanding analytic continuation in terms of Laurent series

h determines at each point

$h(z) = \sum\limits_{n = 0}^{\infty} c_n (z - a)^n$

analytic continuation depends on this fact

one might leave it here, but if one wants to go into it, it is very difficult to understand

basically, is 

$f(x) = \sum\limits_{n} c_n e^{inx}$

is this expansion unique? This was answered in the affirmative by Cantor 1870

$\sum c_k \varphi(n - k) - \varphi(x) c_n$