1. Let f:U®\Bbb C be a continuous function on the open subset U of \Bbb C and suppose f has a primitive F on U. Let g:[a,b]®\C be a contour in U such that g(a) = z₁ and g(b) = z₂. Show that

Assuming Cauchy's Theorem for a triangular contour, show that if g:W®\C is analytic on the open, convex subset W of \C and g is any closed contour in \C, then

2. Let f be analytic on an open subset U of \C except for a pole of order p at z₀ Î U.

• Show that, in any neighbourhood of z₀ included in U,

  f(z) = g(z) (z-z0)p

  where g is analytic in the neighbourhood, and g(z₀) = c > 0.
• Using the continuity of g at z₀, show that there exists a d > 0 such that

  |g(z)| > 1 2 |c|

  for all |z-z₀| < d.
• Deduce that

  lim z® z0  f(z) = ¥.

• By letting z tend to 0 in an appropriate fashion, show that the origin is not a pole of f(z) = e^1/z.

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

Show also that, if |f(z)| £ M(R) on g, then

Now suppose that f is analytic on all of \C, and suppose that, for all z Î \C, |f(z)| £ A|z|^k, where A,k are positive constants. Prove that f(z) is a polynomial of degree not exceeding k.

[Cauchy's Integral Formula may be used without proof, but must be clearly stated.]

4. Show that if a function f:\C®\C is differentiable at z Î \C, then f satisfies the Cauchy-Riemann equations at z.

What other restrictions, apart from the Cauchy-Riemann equations, are required to give conditions sufficient for differentiability?

Explain why the Cauchy-Riemann equations are required for differentiability of functions on \C but not for functions on \Bbb R².

Find the analytic function f = u+iv for which

5. (a) Show, by contour integration, that

(b) Evaluate