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 $f \equiv 0$?



A question about entire functions of order 1
The answer is no: think of $f(z):=\frac {\sin z} z$. Indeed,
$$\frac {\sin (t+ic)} {t+ic}=\frac {\sin(t)\cos(ic)+\cos(t)\sin(ic)} {t+ic}. $$


