1. ## Singularities and integrating

Hi, I have some past papers with no answers and not much idea on how to find them, any help appreciated!!
For part a) is it something about$\displaystyle z^4$ lies on the circle with radius $\displaystyle R^4$and so its minimal distance to -sqrt(2)a i$\displaystyle s R^2-4a^4$?
part b) use the M-L lemma to show int(f)$\displaystyle <$$\displaystyle \Pi R z e^iz$/$\displaystyle R^4-4a^4$ which tends to zero as $\displaystyle R$ tends to infinity?
c) singularities would be $\displaystyle z^4=-4a^4$though not sure what these are . Think I may have to brake $\displaystyle z^4+4a^4$down into composite parts e.g.$\displaystyle (z^2-2ia^2)(z^2+2ia^2)$
part d) I have no idea!

2. Hint For R sufficiently large we have

$\displaystyle \left |{\dfrac{1}{z^4+4a^4}}\right |\leq \dfrac{1}{R^4-4a^4}$

By a well known property

$\displaystyle \left |{\displaystyle\int_{\Gamma_R}f(z)dz}\right |\leq \dfrac{\pi R}{R^4-4a^4}$

Now, take limits in both sides as $\displaystyle R\to +\infty$ .