Further Analysis | Part IB, 2004

(i) State and prove Rouché's theorem.

[You may assume the principle of the argument.]

(ii) Let 0<c<10<c<1. Prove that the polynomial p(z)=z3+icz+8p(z)=z^{3}+i c z+8 has three roots with modulus less than 3. Prove that one root α\alpha satisfies Reα>0,Imα>0\operatorname{Re} \alpha>0, \operatorname{Im} \alpha>0; another, β\beta, satisfies Reβ>0\operatorname{Re} \beta>0, Im β<0\beta<0; and the third, γ\gamma, has Reγ<0\operatorname{Re} \gamma<0.

(iii) For sufficiently small cc, prove that Imγ>0\operatorname{Im} \gamma>0.

[You may use results from the course if you state them precisely.]

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