Problem sheet 4 Feb 1st 2005

MT290 Complex variable

Ex. 1 Transform each of the following equations into a quadratic equation in e^{z} and hence find all the solutions for z ∈ ℂ to

(a) cosh z = −1; (b) sinh z = \frac{i\sqrt{3}}{2}.

Ex. 2 Let f(x + iy) = u(x, y) + iv(x, y). Suppose that f is differentiable as a function of a complex variable, and that v(x, y) = (u(x, y))^{2}. Use the Cauchy-Riemann equations to show that

\frac{∂u}{∂x} = −4u^{2} \frac{∂u}{∂x}

and thus deduce that f(z) is constant. (Note: 1 + 4u^{2} ≠ 0 !)

Ex. 3 By writing tan z in terms of w = e^{iz} write down the quadratic equation w must satisfy if

tan z = a.

Hence find z if tan z = \sqrt{3} − 2i. By considering the quadratic equation which w satisfies, and using the fact that e^{z} = 0 has no solution, show that tan z takes all values in ℂ with two exceptions.

Ex. 4 Sketch the following sets and investigate whether they are open, and/or connected (for (a)-(c) you will want to write z = x + iy):

(a) {z : Im z^{2} > 1}; (b) {z : Re z^{2} ≤ 1};

(c) \{z : Re z^{2} > 1, x > 1, y^{2} < 4\}; (d) \{z : arg z = \frac{π}{6} , 0 < |z| < 1\}.

Ex. 5 Sketch and investigate the following contours to see whether or not they are smooth, piecewise smooth, simple, closed: (a) ϕ(t) = t + i|t − 1|, 0 ≤ t ≤ 2 (b) ϕ(t) = 3 sin t + 4i cos t, 0 ≤ t ≤ 2π

(c) ϕ(t) = sin t + i sin 2t, 0 ≤ t ≤ 2π.

For the two problems above compare your earlier sketches on problem sheet 2.

Ex. 6 Let z = x + iy, x > 0, y > 0. Express log |z| and arg z as functions of x and y (so don’t use i: log |z| = log |x + iy|). Thus show that the function f(z) = log |z| + i arg z satisfies the Cauchy-Riemann equations for x, y > 0. Since the derivatives are continuous in this region this implies that f(z) is differentiable. Show that f′(z) = \frac{1}{z}.