Problems to understand complex infinity limits.

Question

I was thinking a problem like this:

Imagine you have an entire function $f(z)$. Then u can write $f(z)=\sum_{n\ge0}a_nz^n$ for Taylor; Wich is a serie. Then i consider $\lim_{z \to \infty}f(z)$. And now i have multiple questions:

a) Does it always exists?

b) If there exists an $n_0$ that $a_n=0$ $\forall n\ge n_0$. It exists too?

I have a result wich say that if we have an entire function and $\lim_{z \to \infty}f(z)=\infty$ then necessary it's a polynominal. I just want to understand theory of complex funcionts limits.

Answer 1

First of all, let us be clear: the definition of limit at $\infty$ that I work with in the context of complex analysis is this:

• $\lim_{z\to\infty}f(z)=w\in\mathbb C$ if $(\forall\varepsilon>0)(\exists M\in\mathbb{R}^+):|z|>M\implies|f(z)-w|<\varepsilon$;
• $\lim_{z\to\infty}f(z)=\infty$ if $(\forall M\in\mathbb{R}^+)(\exists N\in\mathbb{R}^+):|z|>M\implies|f(z)|>N$.

The limit $\lim_{z\to\infty}f(z)$ doesn't have to exist. It doesn't exist if $f$ is, say, the exponential function, the sine function or the cosine function. On the other and, if $a_n=0$ if $n$ is large enough, then the limit exists, and it is equal to $\infty$. Finally, this works in the opposite direction too: if $f$ is entire and $\lim_{z\to\infty}f(z)=\infty$, then $f$ is polynomial. That's a consequence of the Casoratti-Weierstrass polynomial.