Problem sheet 11 March 22nd, 2005

MT290 Complex variable

Ex. 1 Evaluate the integral ∫_{|z|=1} \frac{e^{−z}}{z^{2}} dz (taken anticlockwise)

a) by means of Cauchy’s integral formulae

b) by evaluating the residue Do the same for ∫_{|z|=1} \frac{sin(2z)}{z^{2}} dz

∫_{|z|=1} \frac{sin(z)}{z^{5}} dz ∫_{|z|=1} \frac{cosh z}{z^{2}} dz ∫_{|z|=1} \frac{cosh z}{z^{3}} dz.

Ex. 2

Note: some of the residues have been calculated on the previous example sheet! 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. 3 Calculate the integral

∫_{C} \frac{1}{e^{z} + 1} dz

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

Ex. 4 Prove (using Complex variable techniques) that

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

Clearly state those results from the lecture that you use.

I hope you find some time for preparing your exams. I intend to give a general revision class about a week before the exam. Based on your questions! I updated the webpage. All problem sheets and the mathematica notebook (on the Fundamental Theorem) are online. http://www.ma.rhul.ac.uk/∼elsholtz/WWW/lectures/0405mt290/lecture.html Happy Easter vacations! I hope you find some time for preparing your exams. I intend to give a general revision class about a week before the exam. Based on your questions! I updated the webpage. All problem sheets and the mathematica notebook (on the Fundamental Theorem) are online. http://www.ma.rhul.ac.uk/∼elsholtz/WWW/lectures/0405mt290/lecture.html