Complex Analysis Homework

 

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Homework 15 – Due 12/12/2018

I. Read Sections 93-94

II. Let C be the curve |z|=2 oriented positively. For each of the following functions f(z), determine how many times (with sign) the image f(C) winds around the origin:

1.     f(z)=z3

2.     f(z)=z4/(z-1)2

3.     f(z)=1/(z2+1)2

III. Do 94.5, 6c, 7a, 9

 

Homework 14 – Due 12/5/2018

Part A

I. Read Sections 80-81

II. Do 81.1ac, 2ab, 3b, 5ab,

III. Do 83.11

Part B

I. Read 84-86

II. Do 86.1, 2, 5, 9

 

 

Homework 13 – Due 11/28/2018

Part A

I. Read Sections 82-83

II. Do 83.1, 2, 4a, 5a

 

 

Homework 12 – Due 11/19/2018

Part A

I. Skim Sections 69-72

II. Do 72.1, 4, 6, 7 (you might want to make different arguments in different parts of the domain)

Part B

I. Read Sections 74-76

II. Show that if f is analytic in a domain except for finitely many singular points then they are all isolated singularities.

III. Do 77.1abc, 2ad

IV. Let C be the positively oriented circle |z|=5. Compute the integral around C of

(sin z)/(z-π)2.

Part C

I. Read Sections 77-79

II. Do 77.3, 4ab, 7

III. Do 79.1abc, 2b

 

Homework 11 – Due 11/7/2018

Part A

I. Read 62-64

II. Do 65.2b, 4, 8a, 9

III. Find a Maclaurin series for z3/(z2+16). On what set does this converge?

IV. Find a Taylor series for z/(1-z) centered at z=3. What is the region of convergence?

Part B

I. Read 65-68

II. Do 65.10, 11

III. Find Laurent expansions about z0=0 for 1/(z3-4z)$ on the regions 0<|z|<2 and |z|>2.

IV. Do 68.5 just the D2 part.

V. Let C be the contour |z|=2 oriented positively. Compute the integral of f(z)=z cos(1/z) around C.

 

 

Homework 10 – Due 10/31/18

Part A

I. Read 54-57

II. Do 57.1abce, 2a, 3, 5, 7, 10

Part B

I. Read 58-61

II. Do 59.1, 3 (this shows that I said something wrong in class), 8

III. Suppose f(z) is entire and |f(z)|≥1 for all z. Show that f is constant.

IV. Let R be a closed bounded region of the plane. Suppose f and g are continuous on R and analytic in the interior of R. Show that if f=g on the boundary of R then f=g on all of R.

V. What is the maximum of |exp(iz2)| on the disk |z|≤1.

VI. Do 61.1

VII. Show directly from the definitions that if  and  then  

 

 

 

Homework 9 – Due 10/24/18

I. Read 50-53

II. Do 53.1acf, 2a, 3, 6, 7

 

Homework 8 – Due 10/17/18

I. Read 47-49

II. Do 43.5

III. Do 47.1a, 2, 4, 7

IV. Do 49.1, 2b, 3, 4, 5

 

Homework 7 – Due 10/10/18

Part A

I. Read Sections 34-37, 41-43

II. Do 34.3

III. Do 36.1a, 2ac, 6, 8a

Part B

I. Read 43-46

II. Do 42.2ac, 4

III. Do 43.1a

IV. Do 46.1a, 2b, 4, 5, 6

 

Homework 6 – Due 10/3/18

Part A

I. Read Sections 30, 37, 38

II. Do 30.1b, 3, 6, 8a, 11

III. Do 38.7 for |sin z|2

IV. Show that sin2z+cos2z=1

V. Show that

VI. Show that   and  

Part B

I. Read Sections 31-33

II. Do 33.1a, 2c, 4, 9, 10a, 11

 

 

Homework 5- Due 9/26/18 (or earlier if you want it graded before the test)

I. Read Sections 25-27

II. Do 26.1.c, 2.a, 4.a, 6, 7

III. Find all real values of a,b,c,d so that ax3+bx2y+cxy2+dy3 is harmonic?

IV. Show by hand that u= x3-3x2y-3xy2+y3 is harmonic and then find a v so that f=u+iv is entire (hint: use the Cauchy-Riemann equations).] (Such a v is called a harmonic conjugate of u).

 

 

Homework 4

Due 9/19/18

Part A

I. Read Sections 17-20

II. Show that the limit as z goes to z0 of f(z) is w0 if and only if the limit as z goes to z0 of f(z)-w0 is 0.

III. Do 18.10, 13

IV. Do 20.1, 8a, 9

Part B

I. Read Section 21-24 (Section 24 will not be discussed in detail in class)

II. Do 24.1bcd, 2b, 3ab, 4

 

 

 

Homework 3

Due 9/12/18:

Part A

I. Do Exercises 12.8, 9

II. Do Exercises 14.5, 8

III. Find an equation satisfied by all the complex numbers that are taken to the line x=1 under the map w=z3. Without looking it up or using any algebraic geometry you should be able to give a rough sketch of this set.

Part B

I. Read Sections 15-16

II. Do Section 18 #1a, 1c, 5, 6.b

 

Homework 2

Due 9/5/18:

Part A

I. Read Sections 9-11, 5

II. Do Exercises after Section 9 #1, 10a

III. Do Exercises after Section 11#1, 2, 3, 4

IV. Do Exercises after Section 5#2, 3

Part B

I. Read Section 12, 13, 14

II. Show that a set is closed according to the book definition if and only if it is the complement of an open set.

III. Do Exercises 12.1, 2, 3, 4abd

 

Homework 1

Due 8/29/18:

Part A

I. Read Sections 1-3

II. Do Exercises after Section 2#1.ab, 2, 4, 5, 8 (hint – you can use the corresponding properties of real numbers), 11 (you can write z=x+iy if you’d prefer)

III. Do Exercises after Section 3# 1a, 5

NOTE: If you look through the other exercises in these section, you’ll notice that there are a lot of basic and familiar algebraic properties that one should technically verify for the complex numbers. For example, if we define -1 as the additive inverse of 1, then how do we know that the product (-1)z is the additive inverse of z? These sorts of properties can be shown by hand for the complex numbers. They can also be shown abstractly for things like fields, which is done in Abstract Algebra classes. From now on, we’ll treat all such properties as true without further mention.

Part B

I. Read Sections 4,6-8

II. Do Exercises after Section 5 #1ad,5, 8, 9

III. Do Exercises after Section 6 #1abc, 2, 3 (just show identity (4)), 10

IV. Do Exercises after Section 9 #2