For the function $$f(z)=z^8+e^{-2016\pi z}$$ defined on $\mathrm{Re}(z)>0$, I am tring to find all roots.
I have initally tried finding the roots of the function via Rouché's Theorem. I at first thought this function had no zeros in this region, however, after playing around in Mathematica, I now believe there should be $2024$ distinct roots. I can't think of a function which would satisfy the conditions of Rouché's Theorem and have that many roots, so I don't believe this is probably the best way to go about it.
From this point, I thought perhaps we could use the Argument principle, by it does does not appear the integral will be easy to evaluate. It can be easily shown that all the roots with $\mathrm{Re}(z)>0$ lie within the open half unit disk, so the path around this disk might be best if I went through such an approach.