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?