# Order is Maximum of Exponent of Convergence and Degree

## Theorem

Let $f: \C \to \C$ be an entire function.

Let $\omega$ be its order.

Let $\tau$ be its exponent of convergence.

Let $h$ be the degree of the polynomial in its canonical factorization.

Then:

$\omega = \max \left({\tau, h}\right)$

## Proof

$\tau \le \omega$
$h \le \omega$

Thus:

$\max \left({\tau, h}\right) \le \omega$
$\omega \le \max \left({\tau, h}\right)$

$\blacksquare$