# Applying the argument principle to functions involving e^z

Typically, you can find the number of zeros of a pole-free function by finding the image of a very large circle in the complex plane. How do you do this if the function includes e^z?

On the circle, we have $z=Re^{i\theta}$. If my function is $e^z+z^3+a=f(z)$ then I have: $$e^{Re^{i\theta}} + R^3e^{i3\theta} +a$$ If the second term were the dominant term, then I could find the image fairly easily (approximately a circle of radius $R^3$ which winds around the origin 3 times). But, as far as I can tell, the first term ($e^Re^{e^{i\theta}}$) is dominant, since $e^R >> R^3$ for large R.

I honestly have no idea how to deal with $e^{e^{i\theta}}$. Normally, I would consider $e^{x+iy}$, but in this form, the relation between x and iy seems less useful.

How do you find the image of the circle?

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We can use Rouchés theorem to show that $e^z + z^3+ a$ has the same number of zeros as $z^3 +a$.
We can then proceed to find the image of that large circle of $z^3+a$. I'm fairly certain this is a curve that winds around the origin 3 times (i think that here we can say that since a is a fixed number, R can be made much larger than a for all a).
If this is correct, then, by the argument principle, since $z^3+a$ has no poles, it has 3 zeros. Thus, by rouchés theorem, our function also has 3 zeros.