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:
- $\omega = \map \max {\tau, h}$
Proof
By Exponent of Convergence is Less Than Order:
- $\tau \le \omega$
By Hadamard Factorization Theorem:
- $h \le \omega$
Thus:
- $\map \max {\tau, h} \le \omega$
By Order is Less Than Maximum of Exponent of Convergence and Degree:
- $\omega \le \map \max {\tau, h}$
$\blacksquare$
Sources
- 1932: A.E. Ingham: The Distribution of Prime Numbers: Chapter $\text {III}$: Further Theory of $\map \zeta s$. Applications: $\S 7$: Integral Functions: Theorem $\text F$