I'm studying for an exam and trying to solve the following:
Suppose that $f(z)$ is a meromorphic function on the extended complex plane (including $z = \infty$) with only two poles: $z = -1$ is a pole of order 1 with $\frac{1}{z+1}$ as the principal part, and $z = 2$ is a pole of order $3$ with $\frac{2}{z-2} + \frac{4}{(z-2)^3}$ as the principal part. Suppose further than $f(0) = 1$.
(a) Determine $\int_{|z| = 4} f(z) dz$.
(b) Determine the number of solutions to $f(z) = 1$ in the extended complex plane.
(c) Determine $f(z)$ explicitly.
I used the Residue Theorem for part (a) to get $6\pi i$ as the solution, but I'm not sure how to attack (b) or (c). My inclination is to apply the Argument Principle to $g(z) = f(z) - 1$, but I'm not sure how to calculate $\int \frac{g'(z)}{g(z)}dz$. As for part (c), I have no ideas yet.
Any help would be appreciated!
A function meromorphic on the whole Riemann sphere is necessarily a rational function. I would try $$f(z)=\frac1{z+1}+\frac2{z-2}+\frac4{(z-2)^3}+h(z)$$ with $h$ a polynomial. The condition $f(0)=1$ will determine $h(1)$.
To avoid a pole at $\infty$, set $w=1/z$. Then $$f(z)=\frac{w}{1+w}+\frac{2w}{1-2w}+\frac{4w^3}{(1-2w)^3}+h(w^{-1}).$$ To avoid a pole at $w=0$, $h$ must be a constant.