Problem sheet 2 Jan 20th 2004

MT290 Complex variable

Ex. 1 Find the derivatives \frac{df}{dz} (with some explanation) and show that the Cauchy-Riemann equations are satisfied for the functions f_{1}(z) = z^{4} and f_{2}(z) = e^{2z}.

Ex. 2 Let f(x + iy) = x^{2} + 2ixy. Writing f = u(x, y) + iv(x, y) find the partial differentials of u and v. Find where the Cauchy-Riemann equations are satisfied and hence find where this function is differentiable (the answer is a simple line).

Ex. 3 Let f(x + iy) = y(1 − x^{2}) − iy^{2}x. Find the set of points where this function is differentiable and sketch the set on an Argand diagram. Do the real and imaginary parts of this function satisfy Laplace’s equation ?

Ex. 4 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| won’t do !). 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 that means that f(z) is differentiable. Show that f′(z) = \frac{1}{z}.

Ex. 5 The function f(x + iy) = u(x, y) + iv(x, y) is differentiable on ℂ, with f(0) = 0, and u(x, y) = e^{x}(x cos y − y sin y). Find v, and hence show that f(z) = ze^{z}.

Ex. 6 (Exam problem 2002)

(i) If f(x + iy) = u(x, y) + iv(x, y) and f(z) is differentiable at z = c, prove that, at c, u and v satisfy the Cauchy-Riemann equations

\frac{∂u}{∂x} = \frac{∂v}{∂y} , \frac{∂u}{∂y} = −\frac{∂v}{∂x}.

(ii) If z = x + iy, write ze^{z} in the form u(x, y) + iv(x, y). Hence verify that ze^{z} satisfies the Cauchy-Riemann equations.

(iii) Let f(z) be differentiable at z, f(0) = 0, and f(z) = u + iv with u = x^{3} − 3xy^{2}. Find f.

(iv) By considering Laplace’s equations, or otherwise, show that if u(x, y) = x^{3} − 2xy^{2} there is no differentiable function f(z) with f(x + iy) = u(x, y) + iv(x, y).

An electronic version of this problem sheet is available at http://www.ma.rhul.ac.uk/∼elsholtz/04mt290/lecture.html Recommanded reading: Murray R. Spiegel: Theory and problems of complex variables. (510.76 Spi). Donald W. Trim: Introduction to Complex analysis and its applications. (515.24 Tri). There are copies of these (and a few other books) on the restricted book shelves. Especially the book by Spiegel has numerous solved exercises.