Answer THREE questions

Indicate briefly how this result can be used to show that if f is analytic on a convex set U then the integral of f round any closed contour in U is equal to zero.

2. Let a Î \Bbb C, R Î \Bbb R, R > 0, and let f be analytic on [`(D)] = {z Î \Bbb C: |z-a| £ R}. Prove that, for all z Î D,

Now consider the power series expansion of f(z) = e^z about the origin. Using the above inequality, what is the best lower bound for n! in terms of n and e which you can obtain?

3. (a) Let f be a function which is analytic over all \Bbb C. Suppose there is a constant M such that |f(z)| £ M for all z Î \Bbb C. Prove that f is constant.

[Any results quoted in Question 2 may be used without proof, if required, but their use must be clearly indicated.]

(b) Let p(z) = a₀+a₁z+...+a_nzⁿ be a polynomial over \Bbb C with degree n ³ 1. Show that, if |z| > 1, then

• Deduce that the polynomial has a root in \Bbb C.
• Let the degree n of the polynomial be at least 2. Consider the integral

  I = ó õ g  1 p(z) dz

  where g is a circle centre at the origin and radius R.

  By finding the limit of I as R®¥, show that the sum of the residues of 1/p(z) at all the roots of p is zero.

4. Show, by means of contour integration, that

5. Let g be a circle with centre at the origin and radius 1 and let a Î \Bbb C. Prove that

Deduce that