Can you help me with this?
For $\lambda \in \mathbb{C}$, show that there exist $m,n \in \mathbb{Z}$ large enough (depending in $\lambda$) such that the equation $e^z = z+\lambda$ has exactly $m+n$ solution in $\{z\mid -2m \pi < \Im(z) < 2n \pi\}$ (where $\Im(z)$ is the imaginary part of $z$).
Thanks a lot! Jon
The idea is to use to use the Argument Theorem and to see what is the change of the arg.