Problem sheet 11 March 23rd, 2004

MT290 Complex variable

Note: some of the residues have been calculated on the previous example sheet !

Ex. 1 Calculate the following two integrals:

∫_{−∞}^{∞} \frac{cos x}{(x^{2} + 1)(x^{2} + 4)} dx, ∫_{−∞}^{∞} \frac{sin x}{(x^{2} + 1)(x^{2} + 4)} dx.

Ex. 2 Calculate the integral

∫_{C} \frac{1}{e^{z} + 1} dz

around the circle C : |z| = 3 and around the circle C : |z| = 4.

Ex. 3 Prove (using Complex variable techniques) that

∫_{−∞}^{∞} \frac{x^{2}}{(x^{2} + 1)(x^{2} + 4)} dx = \frac{π}{3} .