189-249B: Honors Complex Variables

Assignment 1

Due: Friday, January 16.

1. Stein, page 24, exercise 1.

2. Stein, page 25, exercise 2.

3. Write down the 8th roots of unity as complex numbers in exact form. Your final expression should involve only square roots and no trigonometric functions.

4. Stein, page 27, exercise 9.

5. Let $z_1$ and $z_z$ be complex numbers. Show that $$ ||z_1|-|z_2|| \le |z_1+z_2| \le |z_1|+|z_2|.$$ When do equalities hold in each of these inequalities?

6. Using the fundamental theorem of algebra proved in class, show that every irreducible polynomial in $\mathbb R[x]$ is of degree either $1$ or $2$.

7. Let $f(x) = x^n+a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ be a polynomial with complex coefficients satisfying $$ |a_0|+ |a_1| + \cdots + |a_{n-1}| \lt 1.$$ Show that the roots of $f(x)$ all lie in the open unit disc.

8. Consider the cubic equation of the form $x^3-3p x + 2q=0$, where $p$ and $q$ are real parameters and $x$ is the variable.

(a) Show that this cubic equation has three distinct real roots if and only if $p^3 \gt q^2$ i.e., if and only if the complex number $-q+\sqrt{q^2-p^3}$ is not a real number.

(b) Under the hypothesis of (a), let $\alpha_1$, $\alpha_2$ and $\alpha_3$ be the three complex cube roots of $-q+\sqrt{q^2-p^3}$. Show that the roots of the cubic considered in (a) are the three real numbers $$\alpha_1 + \overline{\alpha_1}, \qquad \alpha_2+\overline{\alpha_2}, \qquad \alpha_3 + \overline{\alpha_3}.$$

(This illustrates the striking fact that in Cardano's formulas for solving the general cubic equation in terms of radicals involving the parameters $p$ and $q$ in the equation, one needs to involve complex numbers even when (and in fact, especially when) all three roots of the original cubic are real. This realisation by the algebraists of the Italian Renaissance provided a historical impetus for the study and gradual acceptance of complex numbers as a mathematical tool.)

9. Let $T$ be a general triangle in the plane, with vertices $A$, $B$ and $C$. Recall from your high school geometry days (or from last term's course in history and philosophy of math, for those who took it) that the three angle bissectors in $T$ intersect in a common point $O$ which is the center of the unique circle $S$ inscribed in $T$. Let $r$ denote the radius of $S$, and let $A'$, $B'$ and $C'$ denote the intersection points of $S$ with the segments $BC$, $AC$, and $AB$ respectively. (You may find it helpful at this stage to draw a picture!)

(a) Show that the segments $AB'$ and $AC'$ have a common length, denoted $a$, and likewise that $BA'$ and $BC'$ have a common length denoted $b$ and that $CA'$ and $CB'$ have a common length, denoted $c$.

(b) Show that the complex number $(a+ri)(b+ri)(c+ri)$ is purely imaginary, i.e., that it has real part equal to $0$, and conclude from this that $$ r^2(a+b+c) = abc.$$

(c) Using the formula in (b), prove Héron's formula which asserts that the area of a triangle with side lengths $x$, $y$ and $z$ is equal to $$ {\rm Area}(T) = \sqrt{s(s-x)(s-y)(s-z)}, \qquad \mbox{ where } s = (x+y+z)/2.$$

10. Let $e = 2.7182818...$ be the real constant defined by $$ e = \lim_{N\rightarrow\infty} \left(1+\frac{1}{N}\right)^N.$$

(a) Show that, for any real number $x$, $$ e^x = \lim_{N\rightarrow\infty} \left(1+\frac{x}{N}\right)^N.$$

(b) Let $\theta$ be a real number. Show that the complex numbers $1+ i\frac{\theta}{N}$ and $\cos(\frac{\theta}{N}) + i \sin(\frac{\theta}{N})$ are very close to each other as $N$ gets large, in the precise sense that $$ 1+i\frac{\theta}{N} = \cos\left(\frac{\theta}{N}\right) + i \sin\left(\frac{\theta}{N}\right) + \xi_N, \qquad \mbox{ where } |\xi_N| \lt \frac{C}{N^2}$$ for some constant $C$ depending on $\theta$ but not on $N$.

(c) Use the results in parts (a) and (b) to rigorously justify the loose geometric proof described in class that $$ e^{i\theta} = cos\theta + i \sin\theta.$$