# Math 4283 Fall 2016 Homework

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## Homework 1 (due on Wed. Aug. 24th)

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

## Homework 2 (due on Wed. Aug. 31st)

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 $f(D)$.

## Homework 3 (due on Fri. Sep. 9th)

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?

## Homework 4 (due on Fri. Sep. 16th)

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.

## Homework 5 (due on Wed. Sep. 28th)

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

## Homework 6 (due on Wed. Oct. 5th)

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

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

## Homework 7 (due on Wed. Oct. 12th)

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

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

## Homework 8 (due on Fri. Oct. 21st)

(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

## Homework 9 (due on Fri. Nov. 4th)

23.1, 23.2, 23.7, 23.10, 23.13

24.1, 24.2, 24.5, 24.8, 24.16

## Homework 10 (due on Fri. Nov. 11th)

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

## Homework 11 (due on Mon. Nov. 28th)

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 $\operatorname{Re} s = 1$).