Problem sheet 6 Feb 17th, 2004

MT290 Complex variable

The last problem on this sheet directly corresponds to the lecture from Monday 16th Feb. It may be useful for revision. With problem 4 you will possibly want to wait for Thursday’s lecture. Problems 1,2, and 4 on sheet 5, problem 3 on sheet 4, and problem 5 on sheet 3 are strongly recommended for revision.

Ex. 1 Evaluate the following integrals around C, the circle |z| = 1 taken anticlockwise, using Cauchy’s integral formula (or the formula for derivatives if appropriate).

(a) ∫_{C} \frac{cos z}{z} dz; (b) ∫_{C} \frac{e^{πz}}{(z − \frac{1}{4})^{2}(z − 4)} dz.

Ex. 2 Solve the quadratic equation x^{2} − 4x + 5 = 0. Hence write x^{2} − 4x + 5 = (x−α)(x−β). Use the semicircular contour γ(R): (−R, R)∪{Re^{it} : 0 ≤ t ≤ π} to evaluate the following two integrals:

∫_{−∞}^{∞} \frac{sin x}{x^{2} − 4x + 5} dx, ∫_{−∞}^{∞} \frac{cos x}{x^{2} − 4x + 5} dx.

Ex. 3 Show that

∫_{−∞}^{∞} \frac{cos πx}{x^{2} − 2x + 2} dx = −\frac{π}{e^{π}} , ∫_{−∞}^{∞} \frac{sin πx}{x^{2} − 2x + 2} dx = 0.

Ex. 4 Factorise the polynomial 6x^{2}+13x+6: you should get two factors (ax+b)(bx+a) with a, b both positive integers. Hence evaluate the integral

∫_{0}^{2π} \frac{1}{13 + 12 cos θ} dθ.

Ex. 5 (More difficult) Let

I_{n} = ∫_{0}^{2π} cos^{2n} θ dθ.

(Perhaps you saw somewhere Wallis’s formulae for integrals like these). Show by complex variable techniques that

I_{n} = \frac{π(2n)!}{2^{2n−1}(n!)^{2}} .

Hint. Make the usual substitution to convert the integral to a contour integral around |z| = 1. Use the binomial theorem to expand out the integrand. All the

terms when integrated give zero except the middle one (note that \frac{(2n)!}{(n!)^{2}} is the

coefficient of the middle term in the binomial expansion).

{Ex. 6 \atop (i)}Evaluate in all details ∫_{γ_{1}} z dz, where γ_{1} is the line from z_{1} to z_{2}. and show that ∫_{γ_{2}} z dz = 0, where γ_{2} is the closed quadrangle consisting of the 4 points z_{1}, z_{2}, z_{3}, z_{4}. (Assume for simplicity that the side of the quadrangle do not intersect each other). Now compare with Cauchy’s theorem.

(ii) Convince yourself (with a few less details) that you can similarly do ∫_{γ_{1}} z^{3} dz and ∫_{γ_{2}} z^{3} dz.

(iii) ∫_{γ_{2}} \frac{1}{z} dz, where this time the 4 points all lie on the circle |z| = 2. Explain the difference between (iii) and (iv).

Discuss the orientation. Discuss (using the the deformation of contours theorem) whether it matters or not, that the points are on the circle.

(iv) Now, assuming Cauchy’s theorem and assuming that f satisfies its hypotheses, show that ∫_{γ_{3}} f(z) dz = ∫_{γ_{4}} f(z) dz, where γ_{3} and γ_{4} are any paths from z_{1} to z_{2}. Discuss f(z) = z^{3} and f(z) = \frac{1}{z} as examples.

(v) You know that φ(t) = z = z_{1} +(z_{2} −z_{1})t, 0 ≤ t ≤ 1 describes the line from z_{1} to z_{2}. Which curve is described by: φ(t) = z = z_{1} + (z_{2} − z_{1})t^{3}, 0 ≤ t ≤ 1 ? (Perhaps you will be surprised). Explain your observation. Then evaluate ∫_{γ_{1}} z dz, where γ_{1} is given by this new φ, again and compare the result with (i).

If you had any gaps in the minitest, revise those topics. Revise your knowledge on power (Taylor) series. (MT 172). Revise your knowledge on the convergence of series.

An electronic version of the problem sheets 1-6 and solutions 1-4 is available at http://www.ma.rhul.ac.uk/∼elsholtz/04mt290/lecture.html Please tell me if you find any errors. I slightly edited problem 6(ii) on sheet 2 and the solution to sheet 1, problem 7 and problem 5 on sheet 2.