Further Analysis | Part IB, 2001

State a version of Rouché's Theorem. Find the number of solutions (counted with multiplicity) of the equation


inside the open disc {z:z<2}\{z:|z|<\sqrt{2}\}, for the cases a=13,12a=\frac{1}{3}, 12 and 5 .

[Hint: For the case a=5a=5, you may find it helpful to consider the function (z21)(z\left(z^{2}-1\right)(z- 2) (z3)(z-3).]

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