Vtyjkyuk

This problem was on an old qualifying exam and I was looking for ideas on how to get started:

Let $epsilon>0$, $I= (-epsilon, epsilon) subset mathbbR$ and $gamma: I rightarrow mathbbC$ a curve which is one to one and given by the power series:
$$sum^infty_n=1 a_n(t-t_0)^n$$
that converges in all of $I$
Show that there exists a holomorphic function $Gamma: B(0, epsilon) rightarrow mathbbC$ that agrees with $gamma$ for $t in B(0, epsilon) cap I$

The naive thing to do is to define $Gamma$ to be $$sum^infty_n=1 a_n(z)^n?$$

Analytic continuation guarantees that if we have two holomorphic functions that agree upon a sequence of points that converge to a limit point, then the functions are actually the same function.

In this case we can literally "reuse" the old function, except use a complex variable in its place:

$$sum^infty_n=1 a_n(t-t_0)^n Rightarrow sum^infty_n=1 a_n(z-t_0)^n$$

This function agrees with the original function for all values $z in (-epsilon, epsilon)$, as well as defines a new holomorphic function in the ball $B(0,epsilon)$ and thus adequately extends $gamma$ to $mathbbC$.