Math 106 - Complex Analysis

This course provides an introduction to complex analysis. We will cover complex numbers, holomorphic functions, Cauchy's integral formula, Taylor expansions, meromorphic functions, Laurent expansions, residues and applications.

Instructor: Dragos Oprea, oprea "at" math.you-know-where.edu, Room 382D (2nd floor).

Lectures: Tuesday, Thursday, 11am-12:15pm, Room 381T.

Office hours: Monday 2-4 and 6:30-7:30pm.

Course Assistant: Penka Georgieva, penkag "at" math.you-know-where.edu, Room 381H.

Office hours: Mon 4-6:30, Thr 2:30-4, Fri 3:10-5:10pm.

Textbook: James Brown and Ruel Churchill - Complex Variables and Applications.

There are two copies of the textbook on reserve in the Math library, they circulate for a couple of hours. You can also buy cheaper copies from the internet (check the usual websites).

Prerequisites: Familiarity with multivariable calculus at the level of Math 52 will be assumed.

Exams:

• There will be two in class midterms and a final (December 13).
• The first midterm is in class (Oct 19) and will be graded before the drop deadline.
• The second midterm is a take-home, to be distributed Nov 14, due back Saturday Nov 17. It will be graded before the withdrawal deadline.

Problem Sets: There will be weekly problem sets, usually due on Tuesday in class. The problem sets will be posted online. Group work is encouraged, but you have to hand in your own write up of the homework problems.

• Problem sets (20 percent)
• 2 Midterms (40 percent, 20 percent each)
• Final Exam (40 percent)

Important dates:

• Thanksgiving break: Nov 20-24
• Last day of classes: December 7

Tentative syllabus: PDF

Lecture Summaries

• Lecture 1: Introduction to complex numbers. I showed how to add, multiply, divide complex numbers. The exponential notation, multiplication and powers using complex exponentials.
• Lecture 2: Roots of complex numbers are obtained as vertices of regular n-gons. Distance between two complex numbers in the complex plane is given by the absolute value of the difference. Basic point-set topology: open discs, closed discs, open sets, closed sets, connected sets, bounded sets, compact sets, boundary and closure.
• Lecture 3: Functions of complex variables can be expressed in terms of z and the complex conjugate of z. Functions of a complex variables can be expressed in terms of real and imaginary part of z. Continuity. Definition of continuity using epsilon and delta. Definition of holomorphic functions. I showed that Real part function and the complex conjugate are not holomorphic.
• Lecture 4: Example of a function for which, in order to detect non-holomorphicity, one needs to approach 0 not only from the real and imaginary axes, but also from other directions. I showed that the real and imaginary part of a holomorphic function satisfy the Cauchy Riemann equations. Examples: z bar and z^2. If the real part is known, one can recover the holomorphic function. Examples.
• Lecture 5: Inverting holomorphic functions. Poles. Entire functions. The partial derivatives with respect to z and bar z. I showed that the Cauchy-Riemann equations can be re-written as partial f/partial bar z=0. The converse to the Cauchy-Riemann equations.
• Lecture 6: Harmonic functions and some examples. The real and imaginary part of a holomorphic function are harmonic. I showed that e^{x} cos(y) is harmonic and in fact it is the real part of e^{z} which is holomorphic.
• Lecture 7: Logarithms. log (z) is multivalued, but a holomorphic principal branch Log(z) can be chosen for complex numbers expcept the negative reals. Complex powers z^c are multivalued as well, but that a principal value can be defined as a holomorphic function for z complex but not negative real. For c an integer, the multiple values of z^c all coincide. I computed some examples such as Log(1+i), i^i, (1-i)^i etc.
• Lecture 8: I defined sin, cos, tan, sin^{-1}, cos^{-1} etc by means of the exponential function. I showed sin, cos are holomorphic and computed derivatives. I showed (sin)^2+(cos)^2=1, but that each function is unbounded on the imaginary axis. I computed some examples sin^{-1}(2i), sin (i) etc. Integration. Easy examples via parametrization of curves. Integration of multi-valued functions (such as z^{2i}) when we do or do not cross the branch cut.
• Lecture 9: Integral of any derivative along simple closed curve is 0. The converse is also true, if the integral is 0 then we can find an anti-derivative. Examples: z^2, 1/z^2 along the unit circle. 1/z along the unit circle. I concluded that the integral of dz/z along any positively oriented simple closed loop around the origin is 2pi i. Cauchy's theorem.
• Lecture 10: More on Cauchy's theorem. Simply connected regions. As a corollary of Cauchys theorem and the converse to FTC I showed that every holomorphic function in a simply connected region admits an anti-derivative. For instance, 1/z on the complex plane minus the negative reals admits Log(z) as an anti-derivative. Principle of deformation of paths: integrals of a holomorphic function does not depend on the path if the function is holomorphic IN BETWEEN the two paths. As a corollary, we obtained Cauchy's integral formula.
• Lecture 11: I was away, but Penka covered: connection between Cauchy's integral formula and Cauchy's theorem. Cauchy's integral formula for derivatives. Examples. Liouville's theorem.
• Lecture 12: More on estimates of integrals and Liouville's theorem. Maximum modulus principle. Examples: e^z, sin(z), (z+1)^2 on different compact regions.
• Lecture 13: Power series. Definition, radius of convergence in terms of root test and ratio test. Power series can be differentiated within radius of convergence. All holomorphic functions admit Taylor expansions in the largest disc where they are holomorphic. Examples: e^z, 1/1-z, sin z, cos z.
• Lecture 14: More on Taylor series, showed how Taylor expansion implies the Cauchy integral formula. I worked out expansions around 0 or other points. Stated the theorem about Laurent exapansions of holomorphic functions defined in the region between two circles.
• Lecture 15: More on Laurent expansions. I showed examples such as 1/(1-z)(2-z) which has different expansions in 3 different regions in the plane. I showed that the integral along curves is computed by the coefficient of z^{-1}. I also discussed "division" of power series by means of the example 1/(e^z-1) when 0<|z|<2pi. I started talking about singularities: removable, isolated, non-isolated.
• Lecture 16: The residue theorem. How to compute residues of functions like f(z)/(z-a)^m. The residue theorem implies the Cauchy's integral formula. Easy applications of the residue theorem.
• Lecture 17: Types of singularities: removable, essential, poles. Order of poles. Zeroes and poles. Showed that if f has a zero of order n, g has a zero of order m+n, then f/g has a pole of order m. The residue of f/g when g has a simple zero at a is f(a)/g'(a). Examples.
• Lecture 18: The poles of f'/f are at the zeroes or poles of f. The residues of the poles of f'/f are the multiplicity of the zeroes or poles of f. The argument principle: the integral of f'/f =number of zeroes of f - number of poles of f counted with multiplicity . Rouche's theorem and applications. Started applications to real analysis: evaluation of indefinite integrals.
• Lecture 19: Examples from Fourier analysis via residues.

Homework 1

Due Tuesday, Oct 3, in class. Solve the following problems from the textbook:
• Page 7, Problem 1.
• Page 11, Problem 1(a) and 4.
• Page 13, Problem 2(a).
• Page 14, Problem 14.
• Page 21, Problem 1 and 5.
• Page 28, Problem 2 and 7.

Solutions (written by Penka): Page 1, Page 2, Page 3, Page 4, Page 5, Page 6.

Solutions 2: PDF

Solutions 3: PDF

Midterm 1: PDF

Practice Midterm: PDF

Solutions to Homework 3B: PDF

Solutions to Homework 4: PDF

Solutions to Homework 5: PDF

Midterm 2 due Friday, Nov 17, 4:30pm in my office. PDF

Solutions to Midterm 2: PDF

Solutions to Homework 6: PDF

Practice Final: PDF

Solutions: Page 1, Page 2, Page 3, Page 4, Page 5.

Final Exam: PDF