Determine the multiplicity for the root z = 1 for
$\displaystyle f(z)=(1-cos2\pi z)^{30}$
How exactly?
Well, since $\displaystyle f(z)=g(z)^{30}$, we can write $\displaystyle f(z)=(z-1)^{60}h(z)^{30}$, and since $\displaystyle h(1)^{30}$ we have the order. Hence the power 30 wasn't the main problem, you can try to do the exercise with $\displaystyle f(z)=(1-\cos (2\pi z))^p$ with $\displaystyle p$ a positive integer.