Math 120A

Math 120A, HOME WORK Churchill Ed 8

The homework below will be assigned (likely more too) the due dates far in the future are just approximations.
Please remember to start the problems from each section on a separate page. You should start the HW early, since some of the problems are challenging and the best way to solve them is to think about them over several days. Section is at 7PM on Wednesdays in WLH 2112. Lecture MWF at noon WLH 2115.

Please turn in the HW set at the beginning of the next meeting of section Weds Oct 2
Do not turn in Recommended HW (but do it)

The class text is not yet on reserve in the library. So here are the exercises in Ch 1.. I think it is very slow to download. 25Meg file.
Churchill Ex in Ch 1

chapter one
Sec 2 p5 Ex. 1b , 4,5
Sec 3 p8 Ex. 1a (Recommend 1b, 1c)
Sec 4 p12 (Recommend 5b, 6b)
Sec 5 p14 Ex. 6, 14 (Recommend 7)
Sec 8 p22 Ex 2
Read Supplement to 20B Exercises

Please turn in the HW set at the beginning of the next meeting of section Weds Oct 9

Sec 8 p22 Ex. 6,7
Sec 10 p29 Ex 1, 2, 5, 7, 9
Supplement to 20B Exercises in Sec 1.6 number 1 and 4.

Churchill Sec 11 p.33 1a,b 4a,b
Chapter Two
sec12 p37 1,3,4

Finish the following HW by the beginning of the next meeting of section Weds Oct 16 . You do not need to turn it in because Quiz I is on Friday Oct 18
The quiz contains mostly HW problems you have done with phrasing rearranged, numbers changed, etc. There will in addition be one derivation or proof of something important. For a derivation to be fair game on the Quiz it will be in the book and also have been done in class. The quiz takes all of the class hour.

sec14 p44 3,4,5
sec18 p55 1a, 5, 9
p62 sec 20 complex derivatives 1a, 1b, 2

sec 20 p62 3, 4, 8a,b
p 71 sec 23 1, 2b, 3b (you do not need to read sec 23)
p 77 sec 25 1a 1d, 2 , 3

Week 4

Please turn in the HW set at the beginning of the next meeting of section Weds Oct 23

p 77 sec 25 4a, 4b, 7a

p 81 sec 26 1a 1c , 2, 7
p87 sec 28 1 (read sec 28)

Chapter Three log(x)
p92 sec29 1,2, 5

DATES AND QUANTITY OF EACH ASSIGNMENT WILL BE ADJUSTED BELOW to fit the actual pace of the class.

Week 5

Please turn in the HW set at the beginning of the next meeting of section Weds Oct 30

p87 sec 28 4, 5
p92 sec29 6, 7,8,9,10,11, 14
p96 Ch 3 sec31 1,2a,2b
p96 sec31 3,4,5,10

p100 sec32 1,2,5
p104 sec33 1b, 2a, 6
p108 sec34 1

Chapter Four: Integrals
p 121 sec 38 1 ,2a 2b 2c, 3
p125 sec 39 2,

Week 6:

You do not need to turn HW in because
Quiz II is on Friday , Nov 8.
The quiz contains mostly HW problems you have done with phrasing rearranged, numbers changed, etc. There will in addition be one or two derivations or proofs of something important. For a derivation to be fair game on the Quiz it will be in the book and also have been done in class.

p136 sec42 1, 4, 8, 10
p140 sec43 1, 2, 4
Now what you spend time studying is a little tricky. I suggest downplaying Sec 44 and 45. They do not get used too much. Instead read sec 46 on the Cauchy -Goursat Theorem; it is very important. It gives the classical derivation of the theorem under slightly strong hypotheses. Skip sec 47. It gives the derivation of the theorem under very weak hypotheses.
p161 sec 49 Multiply connected domains: Cauchy -Goursat Theorem exercises 1 , 2

Week 7:

Finish the following HW by Weds , Nov 13

p161 sec 49 More Cauchy -Goursat Theorem exercises 3
p 170 sec 52 1a, 1b, 2a, 3, 5, 7
p 178 sec 54 1, 2 Lioville Theorem

Added Exercise week 7.1
Define the Lapace transform L of a function f:[0, ∞ ] ₊ --> C by
[Lf](z) = ∫ ₀^∞ f(t) e^{- z t} dt
Recall a function G(t) is integrable on [0, ∞ ] means ∫₀^∞ | G(t) | dt exists.
Consider E(t) := e^{b t} where b is a complex number.
(a) Suppose Re b is negative. For which z is the integrand in the Lapace transform of the function E integrable?
(b) Describe the region R of these z on which LE(z) an analytic function of z?
(c) Does LE(z) have an analytic continuation to a much bigger region than R? If so what is the biggest such region?

Added Ex 7.2
Mostly to stimulate your curiousity and maybe a little Extra credit.
Look up the Fourier transform.
Suppose b = -2 . What function is W(r):= LE(i r) ) the Fourier transform of? Here r is a real parameter. Can you say roughly what relationship the Fourier and Laplace transform have?

Week 8:
Midterm on Fri Nov 22 All material in the course up to this point is fair game, with the emphasis being toward the later half of the course material.

This HW set should be done by section Weds Nov 20 dont turn it in since there is a midterm comming

p 178 sec 54 3,4,5, 6 Maximum Principle

Chapter Five: Power Series
You need to read Ch 5 on your own (up to sec 66) more than in previous sections. We will do it quickly in class.
p188 Ch 5 sec56 3,8
p 195 sec 59 2, 7, 13

Thanksgiving Thurs Nov 28 2013 in 9th week of class

Please turn in the HW set the beginning section Weds Nov 27

p 205 sec 62 & Laurent Expansion nbsp Ex 3, 8a
p 219 sec 66 Differentiation of Series, Analytic Continuation 1, 2, 11
Chapter Six:
p 239 sec 71 1a, 1b, 2a, 2c, 3b, 5

p 243 sec 72 1, 2a Thanksgiving Nov 28

Please turn in the HW set the beginning section Weds Dec 5

Week 10
Ch 6
p 248 sec 74 (residue section) ex 1, 3, 7
p 255 sec 76 ex 1,7, 9

The final exam will cover through Ch 6. Just follow the assigned homework for more detail on exactly the sections covered.

SCHEDULE for 2013:
"HW Quiz I" Friday Oct 18 in the 3rd week of class.
"HW Quiz II" Friday Nov 8 during the 6th week of class
Midterm on Fri Nov 22 in the 8th week of class.
Holidays: Veterans Day Mon Nov 11 2013 Thanksgiving Thurs Nov 28 2013 in 9th week of class
Last day of lecture Fri Dec 6 in 10th week of class

Do these over winter break to make your holiday bright:
Ch 7 The goal is to do sec 89 on inverse Laplace Transforms which is way at the end of the chapter. This is section 89 hw. The only section you need to glance at before doing sec 89 HW is sec 81.

Set F(s)= 12/ (s^3 +8) and G(s) = F(s) e^(st).

(1) Set L_3 = vertical line thru 3. What is the principal value integral of G along the line L_3?
(2) Set L_0 = vertical line thru 0. What is the principal value integral of G along the line L_0?

Cultural remark (1) divided by (2 pi i) is the Inverse Laplace Transform of $F$ while (2) is not.