189-249B: Honors Complex Variables

Assignment 3

Due: Friday, February 13.

1. Stein, page 64, Exercise 1.

2. Stein, page 64, Exercise 2.

3. Stein, page 64, Exercise 3.

4. Green's Theorem (which some of you might have seen in earlier classes, although we are not assuming this!) asserts that if $$ (u(x,y),v(x,y)): \mathbb R^2 \rightarrow \mathbb R^2$$ is a continuous function with continuous partial derivatives, and $R$ is a plane region with boundary $C$ (oriented in the positive direction) then $$ \int_C u(x,y) dx + v(x,y) dy = \int_R \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)dx dy.$$ Show that this implies Cauchy's theorem $$ \int_C f(z) dz = 0$$ for any function $f$ that is holomorphic on $\mathbb C$, where $C$ is the same contour as before but viewed in $\mathbb C$ after identifying $\mathbb C$ with $\mathbb R^2$ in the usual way.

5. Stein, page 66, Exercise 9.

6. Stein, page 66, Exercise 10.

7. Stein, page 66, Exercise 13.

8. Stein, page 103, Exercise 1.

9. Stein, page 103, Exercise 2.

10. Stein, page 103, Exercise 3.