Further Analysis | Part IB, 2003

(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)=z5+zp(z)=z^{5}+z. Find all zz such that z=1|z|=1 and Im(p(z))=0\operatorname{Im}(p(z))=0. Calculate Re(p(z))\operatorname{Re}(p(z)) for each such zz. [It will be helpful to set z=eiθz=e^{i \theta}. You may use the addition formulae sinα+sinβ=2sin(α+β2)cos(αβ2)\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right) and cosα+cosβ=2cos(α+β2)cos(αβ2)\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right).]

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

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

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