MAT 342: Applied Complex Analysis

Stony Brook, Spring 2005
Last update: May 11, 2005

This page will be updated at least once a week. Check for announcements and postings regularly, mostly at the bottom of the page!
Announcement: The final exam will be in the usual room (SB Union 226) on Tuesday, May 17 at 8:00 AM. Note that the exam will start much earlier than the regular classes. Do not be late!

Instructor: Valentina Kiritchenko
Phone: 632-8884.
Webpage: http://www.math.sunysb.edu/~vkiritch/
Classes: TuTh 9:50am-11:10am, SB Union 226
Office Hours: Wednesday 2:00-4:00 pm, Thursday 5:00-6:00 pm and by appointment in Math Tower 3-102

Grader: Anirban Dutta
Webpage: http://www.math.sunysb.edu/~dutta/
Office Hours: Tuesday 1-3 pm, MLC ( Math Tower S-240A ) or by appointment

Multiplication by i

What you have to learn for the final: Complex numbers: how to use polar form to solve equations z^n=a; Complex functions, complex linear functions, linear fractional maps, z^2, the exponential and logarithmic function, how to find the image of simple domains (e.g. upper half plane) under the maps az, 1/z, z^2 and e^z and their compositions (e.g. (z+i)/(z-i) and e^(az)); Differentiable complex functions: Cauchy-Riemann (=Euler-D'Alembert) equations, how to use them to prove that a given function is holomorphic. Integration of complex valued functions over contours in the complex plane, upper bounds for moduli of such integrals, Cauchy residue theorem and its applications to the integration of holomorphic functions, how to integrate a function with a finite number of poles (such as e^z/z^2, z/(z^2+1) or 1/(z^2(e^z-1))), how to compute the residue at a simple (=order one) pole, how to compute the residue at a higher order pole using power series; Rouche's theorem, ow it helps to find the number of zeros of a holomorphic function inside a given domain (e.g. how many zeros does z^5+z+1 have in the right half plane); evaluation of improper integrals via Cauchy residue theorem (e.g. how to compute integrals of x^2/(x^4+1) or cos(x)/(x^2+1) from 0 to infinity). Final problems will be similar to those of the midterms. Be sure that you understand how to solve all midterm problems (see solutions below).
Solutions
Picture for Bonus problem 6 from Homework 3 (taken from MathWorld). Steiner porism
Week by week details

About this Course: This is an advanced undergraduate course covering fundamental aspects of functions in one complex variable. We start with the definition and properties of complex numbers including their geometric interpretation. Then we define the notion of complex differentiable function (aka holomorphic) in one complex variable and study some basic examples. We continue with Cauchy integral formula and derive some striking properties of holomorphic functions such as analyticity. We also study isolated singular points of holomorphic functions (Laurent series, residues, Cauchy's Residue formula). Then we apply residues to evaluation of improper integrals. We also cover such topics as harmonic functions and conformal maps.

Prerequisites: MAT 203 or MAT 205 or AMS 261; MAT 200 or permission of the instructor.
Text: James W. Brown and Ruel V. Churchill. Complex Variables and Applications. McGraw-Hill, 7th edition

Additional reading: if you are intereseted in some topics that are not discussed in detail in our course (such as Riemann mapping theorem) you may find useful the following books:
Ahlfors, Lars V. Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York, 1978.
Evgrafov, M. A. Analytic functions. Dover Publications, Inc., New York, 1978.
Both books can be found in Math/Physics Library.

Grading:   There will be 2 tests, 80 minutes each; all given in class - no makeups. If one short exam is missed because of a serious (documented) illness or emergency, the semester grade will be determined based on the balance of the work in the course. A final examination will be held on Tuesday May 17 at 8:00am-10:30. Students are expected to ensure when they register for this course that they will be available for the final examination, and that they do not have too many final exams on that date. The final course grades in MAT 342 will be determined as follows:

Homework/Class Participation 30%, Tests 20% each, Final Exam 30%

Homework/Class Participation: You can not learn mathematics without doing mathematics. It is essential to be an active participant in class and to solve problems.
Each week a homework assignment will be posted further down on this page, normally on Wednesday. It is due the following week by Thursday noon in class or directly with our grader. While you may work together with others in the class (which can be a rewarding experience), write up your own solutions in your own words. Since homework earns credit, it is assumed that everyone submitting particular problems has solved them individually. The goal of the homework is to understand the material, not to merely hand in some paper. This is a more advanced math course where coherent arguments and rigorous proofs are often required. Late homework will not be accepted.
Approximate Course Schedule:
Weeks Sections
January 24-February 24 1-24, 28-30, 83-86
March 1, Tuesday Preparation for the Midterm Test
March 3, Thursday Midterm Test
March 8-April 14 36-50,62,63,69
April 19, Tuesday Preparation for the Midterm Test
April 21, Thursday Midterm Test
April 26-May 5 51-54,61,66,67,69,71,72,74,79,80
May 17, Tuesday, 8:00 am Final exam
Special Needs: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.
Students requiring emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information, go to the following web site.
http://www.ehs.stonybrook.edu/fire/disabilities.asp
Week by Week Details and Homework Assignments:

January 25,27
Chapter 1, Sections 1-9; Definition of complex numbers, arithmetic operations, geometric interpretation (exponential (= polar) form), roots of complex numbers.
Homework 1 (due February 3): pdf,ps,dvi

February 1,3
Review Sections 4-9 Chapter 1; Study Section 83, Chapter 8 and Section 10, Chapter 1; More about geometric interpretation of addition and multiplication of complex numbers (relation to rigid motions and dilations of the complex plane). Curves, domains and regions in the complex plane.
Homework 2 (due February 10): pdf,ps,dvi
Solutions

February 8,10
Review Section 10 Chapter 1; Study Sections 11-13 Chapter 2, Section 29, Chapter 3 and Sections 84-86, Chapter 8; Examples of domains in the complex plane, functions of a complex variable with examples, images of curves and domains under the maps by the exponential function and by the functions f(z)=z^2 and f(z)=1/z, the logarithmic function and linear fractional transformations.
Homework 3 (due February 17): pdf,ps,dvi
Solutions

February 15,17
Study Sections 14-17 Chapter 2; Limits, extended complex plane (=Riemann sphere), continuous and singular points of functions with examples.
Homework 4 (due February 24): pdf,ps,dvi
Solutions

February 22,24
Study Sections 18-24 Chapter 2; Complex derivatives, the difference between complex and real differentiable maps from the complex plane to itself and Euler-d'Alembert (= Cauchy-Riemann) equations, holomorphic functions and conformal maps, Riemann Mapping Theorem (without proof).
No homework. Prepare for the midterm.

March 1,2,3
Review sessions and midterm.
What you should have learnt for the midterm: Complex numbers: how to add, multiply and divide them, how to use polar form to solve equations z^n=a; Domains: how to draw the domain given by an equation, how to find the boundary of a domain; Complex functions, complex linear functions and their geometric interpretation (rotations, dilations, etc.), linear fractional maps, z^2, the exponential and logarithmic function, how to draw the images of simple curves (e.g. lines or circles) under the maps 1/z, z^2 and e^z; Differentiable complex functions: how to find the limit of a function, continuous functions, Cauchy-Riemann (=Euler-D'Alembert) equations, how to find derivatives of z^2, e^z and 1/z. Midterm problems will be similar to those of the homeworks. Make sure that you understand how to solve all homework problems.
Solutions to the first midterm


Homework 5 (due March 10): pdf,ps,dvi
Solutions

March 8,10
Study Sections 36-41 Chapter 4; Integrals of complex valued functions of the complex variable along contours, upper bounds for moduli of such integrals.
Homework 6 (due March 17): pdf,ps,dvi
Solutions

March 15,17
Study Sections 42-44,46,47 Chapter 4; Antiderivatives, Caushy-Goursat theorem, Cauchy integral forula.
Homework 7 (due March 31): pdf,ps,dvi
Solutions

March 29,31
Study Sections 45,48 Chapter 4; Proof of Caushy-Goursat theorem and its applications, higher derivatives of holomorphic functions, Morera theorem.
Homework 8 (due April 7): pdf,ps,dvi
Solutions

April 5,7
Study Sections 49-50 Chapter 4 and Sections 62,69 Chapter 6; Liouville's theorem, fundamental theorem of algebra, maximum modulus principle, residues, computation of the residue at a simple pole.
Homework 9 (due April 14): pdf,ps,dvi
Solutions

April 12,14
Study Section 63 Chapter 6 and Sections 51-54 Chapter 5; Cauchy residue theorem, computation of integrals using Cauchy residue theorem, power series, Taylor expansion formula.
No homework: Prepare for the midterm

April 19,21
Review session and midterm.
What you should have learnt for the midterm: Integration of complex valued functions over contours in the complex plane, upper bounds for moduli of such integrals, how to integrate functions z^n, 1/z^n and z conjugate over simple contours (such as circles or squares), Cauchy-Goursat and Cauchy residue theorems and their applications to the integration of holomorphic functions, how to integrate a function with a finite number of simple (=order one) poles (such as e^z/z or z/(z^2+1)), how to compute the residue at a simple pole, Morera theorem (how it helps to prove that a function is holomorphic), Liouville theorem, Cauchy integral formula for a holomorphic function and its derivatives, maximum modulus principle (how it can be used to show that a holomorphic function has zeros in a given domain). Midterm problems will be similar to those of the homeworks. Make sure that you understand how to solve all homework problems.
Solutions to the second midterm

April 26,28
Study Section 61, Chapter 5, Sections 66,67 Chapter 6 and Section 79 Chapter 7; review Sections 53-54 Chapter 5; Why the Taylor series of a holomorphic function converges to the function (i.e. holomorphic=analytic), computation of residues via Taylor series, argument principle.
No regular homework: to learn better the argument principle solve problems 1 and 2 from Project 2 below.

May 3,5
Study Sections 71,72,74,80 Chapter 7; Rouche's Theorem, evaluation of improper integrals via residues.
No regular homework: to learn better how to compute improper integrals solve problems 1 and 3 from Project 3 below.

Optional Projects: You may submit work on several of the following independent study type problems each worth extra credit up to 10% of the course grade. For some problems it may be helpful to consult not only the textbook but also additional reading sources (see above). More challenging projects are marked by an asterisk. All presentations should be reasonably detailed and neatly typed. Formulas can be entered by hand, if necessary. On the average, an optimal treatment of a topic would probably take 2 pages. Keep it short, but complete. Projects will be added through April. Of course, partial solutions will earn partial credit. Good luck!

Project 1* Conformal mappings
Project 2 Applications of residues (series and zeros)
Project 3 Applications of residues (integrals)