Do check the homework page periodically for new homework or changed homework. Hitting the "Reload" or "Refresh" button on your browser with Ctrl or Shift pressed down (depends on the browser) usually makes absolutely sure that you have the newest version.

Page 9: 1.12, 1.15, 1.17, 1.21

Page 23: 2.10, 2.11, 2.15 (with $z=1+i$), 2.19

Exercise X.1:

a) Convert $2-2i$ to polar form.

b) Use part a) to compute $(2-2i)^{12}$ to polar form.

c) Express $(2-2i)^{12}$ in cartesian form, that is as $x+iy$.

Page 37 (look at the bottom of the page 37!): 3.1, 3.2, 3.5, 3.14, 3.15

Page 45: 4.1, 4.3, 4.4, 4.6, 4.9, 4.10, 4.12, 4.14, 4.16

Exercise X.2:

Let $D$ be a region defined by $D = \{ z \in {\mathbb C} | -\pi/4 < \arg z < \pi/4 \text{ and } |z| < 2 \}$. Let $f(z) = z^2$. Describe and draw $D$ and its image $f(D)$.

Page 53: 5.6, 5.7

Page 60: 6.1, 6.3, 6.4

Page 65: 7.1, 7.2, 7.5, 7.13

Exercise X.3:

Suppose $f(z) = 1/z$, $h(z)={|z|^2}$, $g(z) = \bar{z}$, $\varphi(z) = f(z) h(z)$, and $\alpha(z) = g(\varphi(z))$

Decide which of the functions $f$, $h$, $g$, $\varphi$, $\alpha$
are analytic (wherever they are defined), and if they are analytic compute
their complex derivative.

Exercise X.4:

Let $z =x+iy$.
a) Write $f(x,y) = x^2 - xy + iy$ as a polynomial in $z$ and $\bar{z}$.
b) Is $f$ analytic?

Page 82: 8.1, 8.4, 8.5, 8.8, 8.15, 8.20, 8.29

Page 83: 9.4, 9.5, 9.20

Exercise X.5:

Suppose $u$ is a harmonic function defined on the complex plane. Suppose that
$u_{xx}(x,y) = 2$ for every $(x,y)$. Suppose that $u(0,0) = 0$, $u(1,1) = 0$,
$u(1,0) = 1$, $u(0,1)=3$. Compute $u$. *Hint: Integrate $u_{xx}$ twice in
$x$. Then differentiate twice in $y$ and note what $u_{yy}$ must be, and that
it must be constant (in particular it does not depend on $x$). You will get a
form for $u$ and you solve for all the constants that came up by plugging
in.*

Page 165: 13.1, 13.2, 13.3, 13.7

Page 175: 14.1 (by both line integrals he means both using dx and also using dy), 14.3, 14.5, 14.12

Page 186: 15.2, 15.11

Page 202: 16.1, 16.3, 16.4, 16.10, 16.15

Page 210: 17.1, 17.2, 17.7, 17.11, 17.13

Page 216: 18.6, 18.7, 18.8, 18.9, 18.11

Page 259: 19.2, 19.5, 19.8, 19.9, 19.10

(I only have an electronic version of the book with me now so I don't have the page numbers, you'll have to find the sections)

20.1, 20.3, 20.4, 20.11, 20.15

22.3, 22.4, 22.5, 22.10, 22.13

23.1, 23.2, 23.7, 23.10, 23.13

24.1, 24.2, 24.5, 24.8, 24.16

25.1, 25.2, 25.4, 25.5, 25.6, 25.12, 25.13, 25.16, 25.17

26.1, 26.4, 26.5, 26.6, 26.7, 26.8, 26.9, 26.10

Exercise X.6:

Invert the Laplace transform $F(s) = \frac{s}{(s+1)(s^2+4)}$, that is
compute the integral
$$
f(x) = \frac{1}{2\pi i} \int_{1-i\infty}^{1+i\infty} F(s) e^{-sx} \, ds
$$
as we did in class (that is, use a contour that goes around the poles, so to
the left of the line where $\operatorname{Re} s = 1$).