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I need to prove that $e^z + z$ is a bijection in the left half side of the complex plane.

I thought about showing that the line integral of this function over any curve that starts in $z$ and ends in $w$ is 0 iff $z=w$, thus proving that $e^z + z^2 / 2 =e^w + w^2 / 2$ iff $z=w$.

But this is all I have. I'd be happy with a hint.

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up vote 3 down vote accepted

Perhaps look in your textbook just before this problem?

For example: can you think of a way to show that if your function is not injective in the left half plane, then it must have derivative zero somewhere in the left half plane?

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