Problem sheet 5 Feb 8th 2005

MT290 Complex variable

Ex. 1 Write down in the form ϕ(t), a ≤ t ≤ b the contour C consisting of a circle with 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. 2 Write down in the form ϕ(t), a ≤ t ≤ b the contour consisting of a circle with centre i, radius 3, starting at i − 3 going in an anticlockwise direction. Hence evaluate the integral of z̄ around this contour.

Ex. 3 Let C be the triangular contour joining 0, 1, 1 + i taken in the anticlockwise direction. Sketch C and write down the equations of each of the three lines making up C. Hence evaluate

∫_{C} Re z dz.

Ex. 4 Let γ be the circle with centre 0 and radius r taken anticlockwise. Evaluate (for all r), using the definition of a contour integral,

∫_{γ} z^{n} dz,

where n ∈ ℤ.

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.

Note: Since nobody is coming to the workshops on Friday 1pm it is herewith cancelled. We will try to offer another workshop on Thursdays, possibly 11am, subject to availability of rooms; but anyway it only makes sense if some students are coming! A small number of students is coming on Friday 12pm. Could these please contact me or Dr. Watt: can you also come on Thursdays, 11am or 3pm? This week there will be workshops Thursday 3pm and Friday 12pm.