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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.

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I fixed the math display and the definition of the set to the one which I think you mean (your notation was unclear). Please look it over to see if I correctly interpreted your question. – Willie Wong Jan 18 '11 at 14:07

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