Problem sheet 1 Jan 13th 2004

MT290 Complex variable

Ex. 1 Express the following complex numbers in the form a + bi:

(a) (4 + 3i) + (2 + i), (b) (4 + 3i)(2 + i), (c) \frac{1}{4 + 3i}, (d) \frac{2 + i}{4 + 3i}

Mark the position of each of these numbers on the Argand diagram and write down its conjugate.

Ex. 2 Find the modulus and argument of each of the complex numbers:

a) 1 + i, 1 − i, −1 + i, −1 − i

b) \sqrt{3} + i, −1 + i\sqrt{3}, \sqrt{3} − i, 1 − i\sqrt{3}. Mark the position of each of these complex numbers on the Argand diagram.

Ex. 3 Find the real and imaginary parts of 2e^{−iπ/6} + \sqrt{2}e^{−3iπ/4}.

Ex. 4 Verify that for all complex numbers z, w we have:

\overline{zw} = z̄w̄,  \overline{z + w} = z̄ + w̄,  \overline{zw̄} = z̄w,  |z + w| ≤ |z| + |w|.

(for the last part you could use |z + w|^{2} = (z + w)\overline{(z + w)}, multiply this out, then show that the result is less than (|z| + |w|)^{2}).

Ex. 5 Let z_{1} = r_{1}(cos θ_{1} + i sin θ_{1}), z_{2} = r_{2}(cos θ_{2} + i sin θ_{2}), where r_{1} > 0, r_{2} > 0, and let n be a positive integer. Express each of the following numbers in the form r(cos θ + i sin θ) with r > 0:

(a) z_{1}z_{2}, (b) z_{1}^{n},  (c) \frac{1}{z_{1}} ,  (d) \frac{\sqrt{3} + i}{1 − i},  (e) \frac{i − \sqrt{3}}{−1 − i}

(use your answers to q.2 for (d) and (e)).

Ex. 6 By writing z in modulus-argument form, solve the complex equations:

(a) z^{n} = 1 for n a positive integer, (b) z^{6} = −1,

(c) z^{3} = 1, (d) z^{2} + z + 1 = 0, (e) z^{4} + z^{2} + 1 = 0. Remember:

sin(π/4) = cos(π/4) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2},

sin(π/6) = cos(π/3) = \frac{1}{2}, sin(π/3) = cos(π/6) = \frac{\sqrt{3}}{2}.

Ex. 7 Let f : ℂ → ℂ be a function. Define: f is continuous at c ∈ ℂ. Use the definition to show that f with f(z) = z^{3} is continuous everywhere.

An electronic version of this problem sheet is available at http://www.ma.rhul.ac.uk/∼elsholtz/04mt290/lecture.html