Problem sheet 2 Jan 18th 2005

MT290 Complex variable

Ex. 1 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.

Ex. 2 Simplify

a)  (\frac{1 + \sqrt{3} i}{1 − \sqrt{3} i})^{201} ;  b) (1 + i)^{2n} + (1 − i)^{2n}.

{Ex. 3 \atop a)}Draw in the Argand diagram the sequence of points z^{0}, z^{1}, z^{2}, … for z_{1} = i, z_{2} = exp(\frac{2πi}{12}), z_{3} = 1.1 exp(\frac{2πi}{12}), z_{4} = 0.9 exp(\frac{2πi}{12}).

b) Given z in modulus argument form, give a general formula for z^{n} and for z^{1/n}.

c) For which complex numbers is lim_{n→∞} z^{n} = ∞, and for which lim_{n→∞} z^{n} = 0. For which complex numbers does the limit not exist?

For the specialists: describe as precise as possible what happens, if |z| = 1.

Ex. 4 Sketch the following sets; (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 the following curves:

(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.

(The following problems use material from the 2nd half of the week).

Ex. 6 Find the derivatives \frac{df}{dz} and show that the Cauchy-Riemann equations are satisfied for the functions f_{1}(z) = z^{4} and f_{2}(z) = e^{2z}.

The following two problems are for the specialists (not examinable):

Ex. 7 Here is an alternative construction of the complex numbers.

Let 𝔽 = \{(\begin{matrix}x & y \\ −y & x \end{matrix}) : x, y, ∈ ℝ\}. Show that 𝔽, with the usual matrix addition

and multiplication, is a field. (I.e. show that addition and multiplication satisfies the axioms of a commutative group; check the distributive law.) What is the mutiplicative identity e? Find an element f with f^{2} = −e. Use this to define a bijective map φ : 𝔽 → ℂ which preserves the whole structure of addition and multiplication; i.e. a + b = c in 𝔽 becomes φ(a) + φ(b) = φ(c) in ℂ, etc.

Ex. 8 Now, let ℍ = \{(\begin{matrix}z & w \\ −w̄ & z̄ \end{matrix}) : z, w, ∈ ℂ\}. Show that ℍ, with the usual matrix

addition and multiplication is, well, almost a field. (I.e. show that addition and multiplication satisfies the axioms of a group, but the multiplication is not commutative; check the distributive law.) (In algebra such a structure is called a skew field. The elements of ℍ are called quaternions, they have many interesting properties...

The problem sheets will be available (probably with a few days delay) at http://www.ma.rhul.ac.uk/∼elsholtz/WWW/lectures/0405mt290/lecture.html (The URL given last week was incorrect.) The workshops have plenty of places left! Thursday 3pm, Friday 12, and 1pm.