Problem sheet 4 Feb 3rd 2004

MT290 Complex variable

Ex. 1 Sketch the following sets and investigate whether they are open, and/or connected (for (a)-(c) you will want to write z = x + iy):

(a) {z : Im z^{2} > 1}; (b) {z : Re z^{2} ≤ 1};

(c) \{z : Re z^{2} > 1, x > 1, y^{2} < 4\}; (d) \{z : arg z = \frac{π}{6} , 0 < |z| < 1\}.

Ex. 2 Sketch and investigate the following contours to see whether or not they are smooth, piecewise smooth, simple, closed:

(a) ϕ(t) = t + i|t − 1|, 0 ≤ t ≤ 2

(b) ϕ(t) = 3 sin t + 4i cos t, 0 ≤ t ≤ 2π

(c) ϕ(t) = sin t + i sin 2t, 0 ≤ t ≤ 2π.

Ex. 3 Write down in the form ϕ(t), a ≤ t ≤ b the contour C consisting of a circle centre 1 + 0i, radius 2 starting at 3 + 0i going in an anticlockwise direction. Hence evaluate the following two integrals:

∫_{C} z^{2} dz, ∫_{C} \frac{1}{z − 1} dz.

Ex. 4 Write down in the form ϕ(t), a ≤ t ≤ b the contour consisting of a circle centre i, radius 3 starting at i − 3 going in an anticlockwise direction. Hence evaluate the integral of z̄ around this contour.

Ex. 5 Challenge Question. Let a simple closed smooth contour C be given by ϕ(t), a ≤ t ≤ b (anticlockwise direction as usual !). Let the area enclosed by C be A. Prove that the integral

∫_{C} z̄ dz

is pure imaginary. Show that the integral is 2iA.