I have a question about counting zeros. Here it goes
Given $f(x)= i z^5+z-2010$. Find the number of zeros of $f$ in the upper half plane $\operatorname{Im}(z)>0$.
I have tried to use the Argument Principle as follows:
$$\frac{1}{2\pi i} \int_{\gamma} \frac{f'(z)}{f(z)} =\Sigma n(\gamma,z_i)$$
where $\gamma$ is a large semi disk of radius R Then, I tried to break $\gamma$ into 2 parts, the line [ -R,R] and the upper half circle $R\exp (it), t\in [0,\pi]$ . I can do the integral of the first one and got -1/2 ( as R goes to infinity). However , I do not know to solve for the second integral.