189-249B: Honors Complex Variables

Assignment 4

Due: Friday, March 6.

1. Stein, page 104, Exercise 6.

2. Stein, page 105, Exercise 12.

3. Stein, page 105, Exercise 13.

4. Stein, page 105, Exercise 14.

5. Stein, page 106, Exercise 15 (c).

6. Stein, page 106, Exercise 18.

7. Stein, page 107, Exercise 21 (a)

8. Stein, page 107, Exercise 22.

9. Let $f(z)$ be a polynomial of degree $n$. Show that there is a positive constant $c$ for which $$ \left|\frac{f'(z)}{f(z)} - \frac{n}{z}\right| \le \frac{c}{R^2} \quad \mbox{ for all } z\in C_R(0), $$ once $R$ is large enough. Use this and the argument principle to deduce the fundamental theorem of algebra: the polynomial $f(z)$ has $n$ roots (counted with multiplicity) in the complex plane.

10. Let $R(z)$ be a rational function on $\mathbb C$. (We showed in class that such functions are exactly the functions that are meromorphic everywhere including at $\infty$.)

(a) Show that $$ \sum_{w\in \mathbb C\cup \{\infty\}} {\rm ord}_w (R(z)) = 0.$$

(b) Suppose that $R(z)$ is regular at $\infty$, i.e., that $R(z)$ tends to a non-zero limit at $z\rightarrow \infty$. Show that $$ \sum_{w\in \mathbb C} {\rm res}_w (R'(z)/R(z)) = 0.$$

Remark. This is a special case of the theorem which asserts that if we define $$ {\rm res}_\infty(f(z) dz) := {\rm res}_0 (f(1/z) d(1/z)), \qquad {\rm res}_w (f(z) dz) := {\rm res}_w f(z),$$ for any differential $f(z) dz$ with $f$ a rational function, then $$ \sum_{w\in \mathbb C\cup \{\infty\}} {\rm res}_w(f(z) dz) = 0.$$