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.
Final Grades:
- Problem sets (20 percent)
- 2
Midterms (40
percent, 20
percent each)
- Final Exam (40 percent)
Important dates:
- Drop deadline: Oct 22
- Withdrawal
deadline: Nov
19
- 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