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Complex analysis Demonstration 4 19. 10. 2004

1. Compute by denition the derivatives of (a) z3+z2−z+ 1 (b) z2−1

z2+ 1 (c) (z2−1)(z2−3z).

2. Show that iff0(α) exists, then f(z)is continuous at z =α. 3. Show that each polynomial P(z) is analytic everywhere.

4. Show that the exponential functionf(z) = ez is an entire function (one can suppose that f is analytic at the origin).

5. Show that if f(z) = u(z) +iv(z) is analytic in a region D, and either (a) u, (b) v, (c) |f(z)| is a constant, then f(z) itself must be constant.

6. Show that the following functions are nowhere dierentiable:

(a) f(z) = ¯z (b) f(z) =x.

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