TITLE: The definition of "entire function"
QUESTION [10 upvotes]: I am reading about the definition of "entire functions" :
"If a complex function is analytic at all finite points of the complex plane $\mathbb{C}$, then it is said to be entire ..." 
In fact, I'd like to understand this definition. Thus I wish a help to respond my questions.

Are all analytic functions on $\mathbb{C}$ entire?
Why do we need to use this definition? 

Thank you very much for all of your answers!

REPLY [15 votes]: 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 $\pm i$.