The definition of "entire function" 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!
 A: 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 $\mathbb C$.
A: If $G$ is an open set in $ \mathbb C$, then we write $H(G)$ for the set of all analytic functions $g:G \to \mathbb C$.
A function $f \in H( \mathbb C)$ is called entire.
A: 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 $\pm i$.
