Math 213a: Complex analysis Problem Set #0 (15 September 2003)

A few straightforward exercises in thinking about complex numbers and functions:

1. (i) Let z be the complex number (1 + \sqrt{7} i)/2. Verify that z^{4} = (1 − 3\sqrt{7} i)/2, and

thus find another complex number w such that w^{4} + z^{4} = 1. (ii) Find two complex numbers w and z such that w^{n} + z^{n} + 1 = 0 for every positive

integer n not divisible by 3.

2. (i) Give necessary and sufficient conditions on complex numbers z_{1}, z_{2}, z_{3} to be vertices of an equilateral triangle traversed in the “positive” (counterclockwise) direction. [Hint: the algebra in this exercise will be much easier and more pleasant if you use the R-basis {1, ω} for C, rather than the usual {1, i}, where ω := (−1 + \sqrt{3} i)/2 is a cube root of unity.]

(ii) Let A, B, C be any distinct points in the Euclidean plane, and A′, B′, C′ the points such that triangles A′BC, AB′C, ABC′ are equilateral in the same orientation. Prove that — with what one exception? — the line segments AA′, BB′, CC′ have the same length, and make 60^{∘} angles with each other (extended if necessary).

[It is also known that these three lines are concurrent; when the equilateral triangles are external to △ABC, the point of intersection is known as the Fermat point of △ABC — yes, the same Fermat that Problem 1 should bring to mind. I do not ask that you prove the concurrence, which cannot be easily obtained using the arithmetic of complex numbers.]

3. Let a ≠ 1 be a positive real number. Show that the image of the unit circle (the set of complex numbers z = x + iy such that x^{2} + y^{2} = 1) under the function z ↦ z + a/z is an ellipse. What happens when a = 1, or a is a complex number?

4. Show that the area A enclosed by a simple closed curve C in the complex plane is given by the contour integral

A = \frac{1}{2i} ∮_{C} z̄dz.

(An integral ∫ f dz over some path in the complex plane is interpreted as the line integral ∫ f dx + i ∫ f dy where x and y are the real and imaginary parts of z = x + iy — imagine that dz = dx + i dy. The integrand z̄ is the complex conjugate x − iy of z = x + iy. We’ll soon have a lot more to say about such integrals.)

This problem set is due Monday, September 22, at the beginning of class.