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 $\omega = \max(\tau, h)$.

Proof
By Exponent of Convergence is Less Than Order, $\tau\leq\omega$.

By Hadamard Factorization Theorem, $h\leq\omega$.

Thus $\max(\tau, h)\leq\omega$.

By Order is Less Than Maximum of Exponent of Convergence and Degree, $\omega\leq\max(\tau, h)$.