The definition of "entire function"

Solution 1:

There are 3 definitions of entire functions, all equivalent :

$f(z) = \sum_{n=0}^\infty a_n z^n$ is entire iff it converges for every $z \in \mathbb{C}$ (see the radius of convergence of power series)
$f$ is entire iff it is everywhere analytic, that is for every $z_0 \in \mathbb{C}$ there is $r > 0$ such that $f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$ for $|z-z_0| < r$.
$f$ is entire iff it is everywhere holomorphic.

The Cauchy integral formula for analytic functions (not difficult) lets us show $2. \implies 1.$ And the Cauchy integral formula for holomorphic functions (harder) lets us show $3.\implies 2.$

Note how this doesn't work when $\mathbb{C}$ is replaced by $\mathbb{R}$ : $\ \ f(x)=\frac{1}{1+x^2}$ is analytic on $\mathbb{R}$ but its Taylor series has a finite radius of convergence because of the singularity at $\pm i$.

Solution 2:

Let $f: \mathbb{C} \to \mathbb{C}$ be a complex function.

Then $f$ is an entire function $\iff f$ can be given by an everywhere convergent power series: $$\displaystyle f: \mathbb{C} \to \mathbb{C}: f \left({z}\right) = \sum_{n \mathop = 0}^\infty a_n z^n; \quad \lim_{n \mathop \to \infty} \sqrt [n] {\left|{a_n}\right|} = 0$$

So if $f$ is entire then this means $f$ is holomorphic on $\mathbb{C}$. It must be analytic at every point of $\mathbb C$. In order for that to be true, the function must be defined at every point of $\mathbb C$.