# Show that f is a non-constant polynomial iff...

• Jun 5th 2009, 08:56 AM
morganfor
Show that f is a non-constant polynomial iff...
If f is an entire function how does one show that f is a non-constant polynomial iff. lim z-> infinity |f(z)| = infinity?
the questions suggests that I look at how the MacLaurin series for f and the Laurent series for g(z) =f(1/z) are related and what kind of singularity g has at 0 but i'm not quite sure where to go with that...
• Jun 6th 2009, 04:58 AM
chisigma
There is a theorem [Liouville's theorem...] that states that every bounded entire function is a constant. A function f(*) is said to be 'bounded' if exists a positive number M such that for all z in C is |f(z)| < M...

Kind regards

$\chi$ $\sigma$
• Jun 6th 2009, 05:59 AM
Rebesques
Quote:

Originally Posted by morganfor
If f is an entire function how does one show that f is a non-constant polynomial iff. lim z-> infinity |f(z)| = infinity?
the questions suggests that I look at how the MacLaurin series for f and the Laurent series for g(z) =f(1/z) are related and what kind of singularity g has at 0 but i'm not quite sure where to go with that...

The limit of f as z->infinity is infinity, so 0 is a pole for g. As a pole it has finite order, say m. Then all coefficients of orders greater than m in the Laurent expansion must be zero. So f is a polynomial of order m.