# 4.II.13E

(a) State the residue theorem and use it to deduce the principle of the argument, in a form that involves winding numbers.

(b) Let $p(z)=z^{5}+z$. Find all $z$ such that $|z|=1$ and $\operatorname{Im}(p(z))=0$. Calculate $\operatorname{Re}(p(z))$ for each such $z$. [It will be helpful to set $z=e^{i \theta}$. You may use the addition formulae $\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$ and $\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$.]

(c) Let $\gamma:[0,2 \pi] \rightarrow \mathbb{C}$ be the closed path $\theta \mapsto e^{i \theta}$. Use your answer to (b) to give a rough sketch of the path $p \circ \gamma$, paying particular attention to where it crosses the real axis.

(d) Hence, or otherwise, determine for every real $t$ the number of $z$ (counted with multiplicity) such that $|z|<1$ and $p(z)=t$. (You need not give rigorous justifications for your calculations.)