Carlson's theorem

Remarks

Condition 1 says that $f$ has exponential type in the half-plane. The proof compares $f$ with $\sin(\pi z)$, which vanishes at the integers, and uses the Phragmén–Lindelöf principle to bound $f(z)/\sin(\pi z)$. The bound $c < \pi$ in condition 2 is sharp: $f(z) = \sin(\pi z)$ satisfies conditions 1 and 3, and $\abs{\sin(\pi i y)} = \abs{\sinh(\pi y)}$, which grows like $e^{\pi \abs{y}}/2$. Applied to a difference, the theorem shows that two functions of this kind that agree at all nonnegative integers are equal.

Going deeper

For a student

The theorem justifies many formulas that are first proved for integer values and then used for complex values. For example, two expressions in $z$ that agree for $z = 0, 1, 2, \dots$ and satisfy the growth conditions agree everywhere. It is used for interpolation of sequences and for questions of uniqueness of Newton series. Like the Paley–Wiener theorem, it shows how the exponential type of an entire function controls its behaviour.