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