Suppose $f:\mathbb C \to \mathbb C$ is an entire function on the complex plane of order $1$.

Additionally, suppose that:

$$ \forall\, c \in \mathbb R, \quad \lim_{t \to \pm \infty} \, f(t+ic) =0.$$

Can one conclude that $f \equiv 0$?


The answer is no: think of $f(z):=\frac {\sin z} z$. Indeed, $$\frac {\sin (t+ic)} {t+ic}=\frac {\sin(t)\cos(ic)+\cos(t)\sin(ic)} {t+ic}. $$


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