Let us define open sector and closed sector in complex plane respectively by $\{z : \alpha<arg(z)<\beta \}$ and $ \{z: \alpha\leq arg(z)\leq \beta \}$, where $ 0<\alpha-\beta\leq 2 \pi$.
Consider the following transcendental functions:
$$ f(z)=e^z \cos z \ \ \text{and} \ \ g(z)=\frac{e^z \cos z}{z}$$
In which sectors, if any, do these functions decay to $0$ as $z \to \infty$ ?
Answer:
If $z<0$, then both the functions decays to $0$, otherwise not.
But I think $z$ can not be negative according to the given condition $ 0<\alpha-\beta\leq 2 \pi$.
So how to solve it.
Help me