Complex Variables – Review Questions

Exercise 1 Let the function f(z) = f(x + iy) be analytic in some open connected set S. It is known that

Ref(z) = xe^{2x} cos 2y − ye^{2x} sin 2y .

Identify the function f (as much as possible). Exercise 2 Find all solutions of the equation Log cosh z = 2z. Exercise 3 Let C be the arc g(t), 0 ≤ t ≤ 1, given by

g(t) = t + it^{3/2}, 0 ≤ t ≤ 1 .

(a) Find the arclength.

(b) Evaluate the integral ∫_{C} z Log(z + 3) dz .

(c) Verify that the bound on the absolute value of the integral in terms of the length of C and a bound on the values of the integrand along the arc is satisfied.

Exercise 4 Evaluate the following integrals:

(a) ∫_{C} f(z) dz , where C is the (positively oriented) arc connecting the point 2\sqrt{3} − 2i to

the point 2\sqrt{3} + 2i along the circle |z| = 4, and f is the branch

f(z) = r^{3/2}e^{3iθ/2}, z = re^{iθ}, r > 0, −π < θ < π,

of the function f(z) = z^{3/2}.

(b) ∫_{C} \frac{e^{πz}dz}{z^{2} − 5πiz − 4π^{2}} , where C is the positively oriented circle of radius 100 around 0. Exercise 5 Prove or disprove:

(a) Let f be continuous in an open region S. If there exists a constant K > 0 such that |∫_{C} f(z) dz | ≤ K for every closed (not necessarily simple) contour C, then

∫_{C} f(z) dz = 0 for every such C, and therefore f is analytic in S.

(b) Liouville’s theorem is false for continuous functions, namely there exist non-constant continuous bounded functions defined on the whole of ℂ.

(c) Log 11 = 10 − \frac{10^{2}}{2} + \frac{10^{3}}{3} − \frac{10^{4}}{4} + ….