Complex Analysis

Homework problems from Brown and Churchill.

1. Due Jan 25: p.4 1,4,5 p.7 1 p.11 4, 5(b) p.13 1,2 p.21 5,6 p.28 1,2,3 p.29 6,8(b) p.31 1,2,4
2. Due Feb 1: p.35 1,2 p.42 1-5 p.53 1,5 p.54 11
   Describe the behaviour of the applet for the map f(z)=z+1/z. In particular, calculate
   1. the side lengths of the square
   2. the radius of the circle (it is not 1)
   3. the two special (singular) points
   Give evidence for (i), (ii), and (iii).
3. Due Feb 8: p.59 1 p.60 4,8 p.68 1,2,3
   1. Find the image of 0< Arg(z) < &theta under the mapping w=z³. How large must &theta be so that the w-plane is covered just once?
   2. Find the image of the strip |Re(z)| < 1 under the following transformations: (a) w=2z+i (b) w=1/z
4. Due Feb 15: p.73 1(a)(c) p.74 2(b),4(b),7 p.78 1(a)(b) p.79 7 p.89 1(b), 2 p.90 8(b) p.94 1(a), 2(b), 3 p.96 2 p.99 1
   If f is analytic, show that log |f| is harmonic in two ways:
   1. differentiate (twice) log |f|=½ log |f|²= ½ log (u²+v²)
   2. show log |f|=Re(log f) and use Cauchy-Riemann.
5. Due Feb 22: p.115 2 p.120 2 p.121 5 p.129 1(a)(b),4,5,6 p.130 10 p.133 1,2 p.134 5 p.142 4
6. Due Mar 8: p.153 1,2 p.156 7 p.162 1(a)(b)(c) p.163 2,5 p.171 1
7. Due Mar 22: p.181 1,2 p.188 1 p.189 2,3,5,6,7 p.190 10,11
8. Due Mar 29: p.198 1,3,4 p.218 1 p.219 3,4 p.230 1
9. Due Apr 12: p.230 2(a)(d),3(a)(b) p.233 1(a)(c)(e) p.238 1(a)(b),2(a)(b),3,5 p.257 1-5
10. Due Apr 19: p.245 1,2 p.265 1,3,5 p.276 2,3 p.277 6
11. Due Apr 26: p.286 6,7,8 p.301 1,2,4 p.305 2 p.306 7,12
    Suppose f(z) is analytic inside the simply connected (no holes) region R with two simple zeros f(z₁)=0=f(z₂).
    Show that the contour integral along C, the boundary of R, is $\int$ _C zf '(z)/f(z) dz = 2&pi i (z₁+z₂).
    (Hint: consider the proof of the identity $\int$ _C f '(z)/f(z) dz = 2&pi i (1+1).)
12. Due May 3: p.317 3 p.329 7 p.350 1,3 p.387 6,7 p.390 17
    Using the applet Interactive Conformal Mapping find special points of the functions 1,3 and 4.
    (Hint: in some cases, the map is not angle-preserving there: f'(z)=0)

Exams.

Links to complex analysis applets.

Course content:

This course is an introduction to complex analysis. Complex analysis is an extremely powerful tool that can be applied in many problems both within pure mathematics and in real world applications. It applies to problems involving only real numbers, such as integration of functions in situations where techniques likes substitution fail. This will appear as contour integration in the course. The deep idea underlying complex analysis is the way in which local information has global consequences. Two important examples of this are contour integration and the maximum principle. In physics and engineering the local information arises in the form of differential equations such as the local acceleration or stress felt by a body and measurable features occur as global information. Applications will be mentioned throughout the course.

Three broader aims of this course are to improve your ability in the following.

1. Problems: ability to calculate.

2. Proofs: learn to reason and to communicate this.

3. Understanding: the deeper mechanisms.

Text

Complex Variables and Applications, 7th edition Brown and Churchill.

Assessment:

• class attendance 10%
• homework 25%
• two in-class exams 15% + 20%
• final exam 30%

Homework will be assigned each lecture and due the Thursday in the following week. A copy of what is assigned will appear on the web. It will be good practice to complete a portion of the assignment after the corresponding lecture. Problems which do not show reasons or work will not be given full credit. Please staple your homework. Late homeworks will not be accepted without a medical excuse (generally this means a note from a clinic). No makeups for exams will be given, and exams will not be excused without a medical reason.

Homework writeups: Homework and problem solutions should be written up clearly - communication skills will determine a large part of your mathematical success in life, and will also determine a large part of your success in this course. Please make it a habit to rewrite homework solutions if they are not entirely in satisfactory final form on first writeup. In arguments involving proofs, you should focus on sentence construction and clarity of arguments in final write-ups. Proofs can create notorious difficulties if they are not written clearly and succinctly.

Consultation on homework: You are permitted and encouraged to consult with other students on home- work problems, but this should be done on a general level of finding the solution of a problem. The final writeup of a problem set must be done by each student individually.