Problem sheet 7 Feb 24th, 2004

MT290 Complex variable

Ex. 1 Revise previous chapters and problem sheets. Do the problems from sheet 6 you haven’t done so far (and hand them in!) For problems 2 and 3 on sheet 6 work with e^{iz}.

Ex. 2

Let I(a, b) = ∫_{0}^{2π} \frac{1}{a + b cos t} dt.

For which (a, b) do you expect that I(a, b) is a real positive number? Fix a = 1 and evaluate I(1, b) for all 0 ≤ b < 1. You will want to show that the denominator of the corresponding rational function has precisely one root inside the unit circle.

Ex. 3 The function f(z) = u(x, y) + iv(x, y) is differentiable for all z = x + iy, and f(0) = 0. Given that u(x, y) = sin x cosh y, show, using the Cauchy-Riemann equations, that f(z) = sin z. Hint: You will need to use results like sin(iy) = i sinh y and remember those old trig identities like sin(A + B) = . . ..

Ex. 4 Let γ be the circle with centre 0 and radius 2, taken anticlockwise. Use Cauchy’s integral formula, or the formula for derivatives, to evaluate the following integrals:

∫_{γ} \frac{z^{2}}{z − i} dz, ∫_{γ} \frac{sin πz}{(z − 1)^{2}} dz.

Ex. 5 (This exercise shall help to understand the estimation lemma.)

i) Let (in real analysis) f(x) = 10 + sin x. Sketch the function. Give a lower and an upper bound on | ∫_{3}^{7} f(x) dx|.

ii) Let C be any semicircle Re^{it} (t_{0} ≤ t ≤ t_{0} + π) around the origin. Give in all details an upper bound on the integral

∫_{C} \frac{z}{z^{3}} dz.

iii) Use the estimation lemma to get (in detail) an upper bound on

∫_{C} \frac{2z + 1}{z^{3} + 1} dz.

Ex. 6 (This problem shows that a) not only semicircle contours help to evaluate integrals along the real line; b) that you can reduce an integral like ∫ e^{−x^{2}} cos ax dx (here with a = 1) to ∫ e^{−x^{2}}, dx. The latter is not really easy, but at least well known so that you can use it here.)

Let f : ℂ → ℂ be the function with f(z) = e^{−z^{2}}. Let R(K) denote the rectangle defined by the four points P_{1} = −K + 0i, P_{2} = +K + 0i, P_{3} = +K + \frac{1}{2}i, P_{4} = −K + \frac{1}{2}i. Let γ_{1} denote the path along the edge connecting P_{1} and P_{2}, Let γ_{2} denote the path along the edge connecting P_{2} and P_{3}, Let γ_{3} denote the path along the edge connecting P_{3} and P_{4}, Let γ_{4} denote the path along the edge connecting P_{4} and P_{1}.

i) Draw the integration contour in the Argand diagram.

ii) Show that ∫_{∂R(K)} f(z) dz = 0.

iii) Show that lim_{K→∞} ∫_{γ_{2}} f(z) dz = 0, and similarly lim_{K→∞} ∫_{γ_{4}} f(z) dz = 0. Use the above results, and (without

proof) the well known result ∫_{−∞}^{∞} e^{−x^{2}} dx = \sqrt{π} to conclude that ∫_{0}^{∞} e^{−x^{2}} cos x dx = \frac{\sqrt{π}}{2e^{1/4}} .