Order is Maximum of Exponent of Convergence and Degree

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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

By Exponent of Convergence is Less Than Order:

$\tau \le \omega$

By Hadamard Factorization Theorem:

$h \le \omega$

Thus:

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

By Order is Less Than Maximum of Exponent of Convergence and Degree:

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

$\blacksquare$


Sources