Analysis II
Math 2302 Spring 2013


MW 3:50-5:05 pm

https://pitt.zoom.us/j/99800879861

Passcode: privoded in the email


Meetings on Mondays will have a form of recitations and we will discuss examples and homework problems.

Meetings on Wednesdays will have a form of office hours where I can answer any of your questions.


Announcements:

Final Exam (pdf) Final Exam (LaTeX file) Due date: May 2, Canvas.

On Wednesday March 3rd we will have another problem solving session instead of "office hours" so do not miss it.


Instructor: Piotr Hajlasz

Office: Small bedroom in my house

Email: hajlasz@pitt.edu

Office hours: By appointment


Course Grade:

Homework (40%) + Two midterm exams (30% + 30%)


Textbook: The course is based on my lecture notes.

Complex Analysis

Fourier Analysis

Prime Number Theorem

The Fourier Analysis part of the course is based on the material on pp.30-37 and 48-63 in Functional Analysis:

Functional Analysis

Some useful books:

S. Lang, Complex analysis. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999. xiv+485 pp. ISBN: 0-387-98592-1

J. Bak and D. J. Newman, Complex analysis . Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1997. x+294 pp. ISBN: 0-387-94756-6

Rudin, Walter Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1

The harmonic analysis part will be partially based on

H. Dym and H. P. McKean, Fourier series and integrals. Probability and Mathematical Statistics, No. 14. Academic Press, New York-London, 2005. x+295 pp.


Homework:

If you do not have a complete solution, do not submit it as you will get negative points for an incomplete solution. All solutions have to be written in LaTeX using a template provided below. The homework must be submitted through Canvas as a pdf file.

HW#1 (pdf) HW#1 (LaTeX file) Due date: February 1, Canvas.

HW#2 (pdf) HW#2 (LaTeX file) Due date: February 8, Canvas.

HW#3 (pdf) HW#3 (LaTeX file) Due date: February 18, Canvas.

HW#4 (pdf) HW#4 (LaTeX file) Due date: February 25, Canvas.

HW#5 (pdf) HW#5 (LaTeX file) Due date: March 11, Canvas.

HW#6 (pdf) HW#6 (LaTeX file) Due date: April 1, Canvas.

HW#7 (pdf) HW#7 (LaTeX file) Due date: April 8, Canvas.


Recorded lectures

January 20

Lecture 1 (Review of complex numbers and complex differentiation.)

Lecture 2 (Cauchy-Riemann equations.)

January 25

Lecture 3 (Cauchy-Riemann equations from a different perspective.)

Lecture 4 (Holomorphic functions and conformal mappings.)

Lecture 5 (Holomorphic functions defined by a power series.)

January 27

Lecture 6 (exp(z), sin z, cos z.)

Lecture 7 (log (z), log'(z)=1/z, power series for log(1-z).)

Lecture 8 (a^b, branch of log(z), branch of z^a)

February 1

Lecture 9 (How to be smooth at infinity. Rational functions are smooth at every point!!!)

February 3

Lecture 10 (Fractional linear transformations I)

Lecture 11 (Fractional linear transformations II)

February 8

Lecture 12 (Line integrals: review of Calculus III in the complex notation)

Lecture 13 (The Cauchy theorem)

February 10

Lecture 14 (Application of the Cauchy theorem: integral of sin x/x)

Lecture 15 (Application of the Cauchy theorem: Fresnel integrals)

Lecture 16 (Application of the Cauchy theorem: Fourier transform)

Lecture 17 (The Cauchy integral formula)

February 15

Lecture 18 (Zeros and uniqueness of holomorphic functions)

Lecture 19 (Uniform limit of holomorphic functions is holomorphic, Cauchy inequality, Liouville theorem, fundamental theorem of algebra)

Lecture 20 (Morera's theorem, Schwatz reflection principle)

February 17

Lecture 21 (Remvable singularities; an enire function that converges to infinity is a polynomial; the only holomorphic functions on the Riemann sphere are rational functions)

February 22

Lecture 22 (Homotopy)

Lecture 23 (Integrals along homotopic curves are equal)

February 24

Lecture 24 (Holomorphic branches in simply connected domains)

Lecture 25 (Harmonic functions, simply connected domains, harmonic conjugate)

March 1

Lecture 26 (Laurent series)

Lecture 27 (Classification of singularities)

March 3

Lecture 28 (Meromorphic functions and rational functions)

Lecture 29 (The Cauchy residue theorem)

Lecture 30 (How to compute the residue I.)

Lecture 31 (How to compute the residue II.)

March 8

Review before the exam

March 10

First midterm exam

March 15

Lecture 32 (Residue theorem: selected applications)

Lecture 33 (Residue theorem: evaluation of integrals I)

March 17

Lecture 34 (Residue theorem: evaluation of integrals II)

Lecture 35 (Residue theorem: evaluation of integrals III)

Lecture 36 (Residue theorem: evaluation of integrals IV)

March 22

Lecture 37 (Residue theorem: evaluation of integrals V)

Lecture 38 (Residue theorem: evaluation of integrals VI)

Lecture 39 (Residue theorem: evaluation of integrals involving branches I)

March 24

(Self-care day: no classes)

March 29

Lecture 40 (Residue theorem: evaluation of integrals involving branches II)

Lecture 41 (not available yet) (Residue theorem: evaluation of integrals involving branches III)

Lecture 42 (not available yet) (Residue at infinity)

March 31

Lecture 43 (Index of a point with respect to a curve)

Lecture 44 (Rouche's theorem, argument principle, Hurwitz's theorem, uniform limits of one-to-one functions)

April 5

Lecture 45 (Applications: finding roots of holomorphic functions I)

Lecture 46 (Applications: finding roots of holomorphic functions II)

Lecture 47 (Applications: finding roots of holomorphic functions III)

Lecture 48 (Applications: finding roots of holomorphic functions IV)

April 7

Lecture 49 (Open mapping theorem)

Lecture 50 (Maximum principle and Schwarz's lemma)

Lecture 51 (Authomorphisms of the unit disc)

April 12

Lecture 52 (Montel's theorem)

Lecture 53 (The Riemann mapping theorem)

April 14

April 19

April 21

April 26

April 28


Recitations 1 (Problem solving session)

Recitations 2 (Problem solving session)

Recitations 3 (Problem solving session)

Recitations 4 (Problem solving session)

Recitations 5 (Problem solving session)

Recitations 6 (Problem solving session)

Recitations 7 (Problem solving session)

Recitations 8 (Problem solving session)

Recitations 9 (Problem solving session)

Recitations 10 (Problem solving session)


Additional notes

The notes below were written when I prepared videos. They are based on the the main lecture notes but sometimes there are more details and some mistakes are fixed.

Cauchy-Riemann equartions

Complex logarithm

How to be smooth at infinity. Rational functions are smooth at every point!!!

Fresnel integrals

Fourier transform

Zeros of analytic functions

Lecture 21

Lectures 43-48

Lectures 52-53


Academic Integrity: Students in this course will be expected to comply with the University of Pittsburgh's Policy on Academic Integrity. Any student suspected of violatingthis obligation for any reason during the semester will be required to participate in the procedural process, initiated at the instructor level, as outlined in the University Guidelines on Academic Integrity. This may include, but is not limited to, the confiscation of the examination of any individual suspected of violating University Policy. Furthermore, no student may bring any unauthorized materials to an exam, including dictionaries and programmable calculators.

Disability Resources: If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services (DRS), 140 William Pitt Union, (412) 648-7890, drsrecep@pitt.edu, (412) 228-5347 for P3 ASL users, as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course. Health and Safety Statement: In the midst of this pandemic, it is extremely important that you abide by public health regulations and University of Pittsburgh health standards and guidelines. While in class, at a minimum this means that you must wear a face covering and comply with physical distancing requirements; other requirements may be added by the University during the semester. These rules have been developed to protect the health and safety of all community members. Failure to comply with them will result in you not being permitted to attend class in person and could result in a Student Conduct violation. For the most up-to-date information and guidance, please visit coronavirus.pitt.edu and check your Pitt email for updates before each class.

Diversity and Inclusion: The University of Pittsburgh does not tolerate any form of discrimination, harassment, or retaliation based on disability, race, color, religion, national origin, ancestry, genetic information, marital status, familial status, sex, age, sexual orientation, veteran status or gender identity or other factors as stated in the University's Title IX policy. The University is committed to taking prompt action to end a hostile environment that interferes with the University's mission. For more information about policies, procedures, and practices, see: Policies, Procedures and Practicies.