NAME Ismael Akray PRACTICAL/TUTORIAL GROUP NAME
Ismael Akray
PRACTICAL/TUTORIAL GROUP
Unit Course book
Complex analysis
2012/2013
Prerequisites. Students with a good understanding of reading and writing proofs can consider taking this course but must obtain permission of the instructor.
Course Description: Complex analysis, the theory of functions of complex numbers, is one of the crowning achievements of nineteenth century mathematics. Although complex numbers are sometimes called imaginary numbers, complex analysis is far from imaginary; it has a multitude of real-world applications to engineering, physics, and applied mathematics.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering.
Murray Spiegel described complex analysis as "one of the most beautiful as well as useful branches of mathematics".
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separable real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and just prior. Important names are Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Traditionally, complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it became very popular through a new boost of complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis today is in string theory which is a conformally invariant quantum field theory.
Complex functions
A complex function is a function in which the independent variable and the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain and range are subsets of the complex plane.
For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts:
and
where
and
are real-valued functions.
In other words, the components of the function f(z),
and
can be interpreted as real-valued functions of the two real variables, x and y.
The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentials, logarithms, and trigonometric functions) into the complex domain.
Holomorphic functions (Analytic functions)
Main article: Analytic function
Holomorphic functions are complex functions defined on an open subset of the complex plane which are differentiable Analytic functions are complex functions defined on a subset of the complex plane which are differentiable . Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic and so analytic.
Major results
One central tool in complex analysis is the line integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary (Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is useful (see methods of contour integration). If a function has a pole or singularity at some point, that is, at that point its values "blow up" and have no finite boundary, then one can compute the function's residue at that pole, and these residues can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by Picard's Theorem. Functions which have only poles but no essential singularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities.
A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed.
An important property of holomorphic functions is that if a function is holomorphic throughout a simply connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions such as the Riemann zeta function which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
All this refers to complex analysis in one variable which is our study for this year. There is also a very rich theory of complex analysis in more than one complex dimension where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) are no longer true. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.
Text Book and Notes: The text book used for the course is Complex Variables and Applications (8th Edition) by James Ward Brown and Ruel V. Churchill, McGraw Hill, 2003. The book can be borrowed through the bookstore of the department.
References: The recommended book for the class is
? J.E. Marsden, M.J. Hoffman, Basic complex analysis, 3rd edition, ISBN 0-7167-2877-X, published by Freeman and Company, 1999.
However, I will not require students to purchase the book. Other good texts are
? Serge Lang, Complex Analysis (especially if you want to go on to graduate school in mathematics)
? E.B. Saff, A.D. Snider, Fundamentals of complex analysis for mathematics, science and engineering (easier text, sometimes not completely mathematically rigorous)
Office Hours: My office is opposite to the math department room, and you can reach me by phone at 009647504497602 or - much preferred - via email at ismaeelhmd@yahoo.com . My office hours are after the lecture, and by appointment. Since the homework will be challenging, it is important that you make appointments with me as soon as any problems arise.
Grading: There will be homework assigned during each class, which will be due the next time class meets. No late homework is accepted, except in special circumstances. There will be two exams during the semester, and possibly a final exam during the officially scheduled time. In addition, each person is required to explain a homework problem on the board at least once. While that performance is not graded, it is required for passing the course. The final grade is computed as follows: 45% homework, 45% exams, 10% participation
Material Covered: The course will cover material that is considered standard for an undergraduate complex analysis course:
? Chapter 1. Complex Numbers (Basic Algebraic, Vectors and Moduli, Conjugates, Exponentials, Products and Powers, Roots, Regions in the Complex Plane)
? Chapter 2. Analytic Functions (Limits, Continuity, Derivatives, Cauchy Riemann Equations, Analytic Functions, Harmonic Functions)
? Chapter 3. Elementary Functions (Exponential, Logarithm, Complex Exponents, Trigs, Hyperbolic Functions)
? Mapping by Elementary Functions
? Chapter 4. Integrals (Definite Integrals, Contour Integrals, Antiderivatives, Cauchy Goursat Theorem, Cauchy Integral Formula, Liouville's Theorem, Fundamental Theorem of Algebra, Maximum Modulus Principle)
? Chapter 5. Series (Sequences, Convergence of Series, Taylor Series, Laurent Series, Absolute and Uniform Convergence, Power Series techniques)
? Chapter 6. Residues and Poles (Residues, Cauchy's Residue Theorem, Residue at Infinity, Zeros of Analytic Functions)
? Additional topics will be included as time permits.
Quizzes: There will also be occasional short quizzes in class.
Homework: There will be about ten homework assignments; the lowest grade will be dropped. You should start working on the homework problems for a section as soon as that section is covered in class. Although you are encouraged to consult with other students and seek help from me, your homework should ultimately represent your own work. Answers unsupported by work will not receive credit. Homework should be neatly handwritten or typed, on one side of the page only. Please remove messy edges and staple.
Tests: There will be two midterms and a final. Makeup tests will only be given to students who contact me within 48 hours of missing a test. Students with a valid, verifiable reason for missing a test may take a makeup without penalty; those who have missed a test without a valid, verifiable reason may take a makeup with a 30% penalty. The final will be cumulative.
Most students are aware of what is allowed during a test. Since tests usually come with specific guidelines, it is easier to determine what is and is not acceptable than it is for out-of-class assignments. Unless an instructor has given special instructions, the following are examples of cheating on a test. (Note that this is not intended as an all-inclusive list):
? Looking at or copying another student’s work
? Allowing another student to copy your work
? Sharing a calculator, other wireless device
? Using a cell phone. (Cell phones must be turned off.)
? Using any materials (notes, books, etc.) not authorized by the instructor
? Discussing a test with students who have not yet taken it, or asking others about a test you have not yet taken. (This includes talking to students from other groups of the same course).
Academic Honesty: Dishonesty includes cheating on a test, falsifying data, misrepresenting the work of others as your own (plagiarism), and helping another student cheat or plagiarize. Academic dishonesty will result in a grade of zero on that particular assignment; serious or repeated infractions of the Academic Honesty policy will result in failure of the course. All violations of the Academic Honesty policy will result in the filing of a violation report
Attendance: If you do miss a class, it is your responsibility to make up the material and make sure your homework is turned in on time.
Schedule:
Weeks
Topics
3
Chapter 1
4
Chapter 2
3
Chapter 3
3
Chapter 4
6
Chapter 5
4
Chapter 6
4
Chapter 7