Week | Sections in Text | Suggested Homework | Comments |
---|---|---|---|
05/01 to 09/01 |
Review of Chapter 3
and 4. Sections 24.1, 24.2, 24.3, 24.4, 24.8 |
3.8 Exercises: 3.10, 3.12, 3.14, 3.16, 3.18,
3.26, 3.28 4.8 Exercises: 4.16, 4.20, 4.23, 4.33, 4.36 |
After a very
brief review on Tuesday of some basic facts about
complex numbers, as contained in Chapter 3, and power
series as contained in Chapter 4, I explained the
exponential finction, its inverse the loagarithm and
why we need a branch cut. After that I defined complex
differentiation and derived the CAUCHY-RIEMANN
EQUATIONS that analytic (or
holomorphic) functions satisfy using the
fundamental fact that the derivative is complex linear
and hence commutes with multiplication by i. I explained
some of the profound consequences of these fundamental
equations that lie at the foundadtion of complex analysis. On Friday, I explained how one integrates
complex functions, or better, one
forms f(z)dz along curves and proved the CAUCHY
INTEGRAL
THEOREM (the
mother of all
theorems in
complex
analysis) as a
consequence of
the C-R
equations and
Green's
Theorem.
I also showed you how to integrate
dz/z around a closed loop,
which is the mother of all
residue calculaions! |
12/01 to 16/01 |
Sections 24.8, 24.9, 24.10, 24.11, 24.12, 24.5, 24.6 | 24.14 Exercises: 24.2, 24.3, 24.4, 24.5, 24.6, 24.7, 24.8, 24.9, 24.10 |
I
continued with the
Cauchy
integral
Theorem,
derived
the
RESIDUE
THEOREM and
discussed
the notion of
the winding
number. The
residue
theorem is the
key technique
(contour
integrals)
that will
be used to
evaluate
(without using
Wolfram
alpha!) a
number of
important
definite
integrals. I also talked about ZEROS, POLES, ESSENTIAL SINGULARITIES, TAYLOR series, LAURENT series, etc. I derived the CAUCHY INTEGRAL FORMULA and its variants (you just differentiate what you have!). This is another fundamental ingredient in complex analysis. |
19/01 to 23/01 |
Sections 24.11, 24.12,
24.13, 25.3 |
24.14 Exercises: 24.11, 24.12, 24.13, 24.14, 24.15, 24.16, 24.17, 24.18, 24.19, 24.20, 24.21, 24.22 |
Assignment
#1 is
due this week on Tuesday at the
beginning of the lecture period. This week was devoted to various techniques of contour integration. We evaluated some classical integrals such as the Fresnel Integral. |
26/01 to 30/01 |
Sections 25.3, 25.4 | 25.9 Exercises: 25.6, 25.7, 25.8, 25.8, 25.9, 25.10, 25.11, 25.12, 25.13, 25.14, 25.15 |
We will also learn
how to use contour integrals to sum interesting infinite series such as Riemann zeta function at even integers (and also at -1) using Bernoulli numbers. That was fun (at least for me!) We explored more consequences of the Cauchy integral formula and the residue theorem, such as the Mean Value Property, the Maximum Modulus Principle, the Argument Principle and LIOUVILLE's THEOREM. . The argument principle can be used to count the number of zeros and poles inside a contour (Rouché's Theorem). The fundamental theorem of algebra follows from that. |
02/02 to 06/02 | Sections 24.7, 25.1, 25.2 | 25.9 Exercises:
25.2, 25.3, 25.4, 25.5 |
Assignment
#2 is
due on Tuesday
at the beginning of the lecture period.
On Tuesday and Thursday, we will talk about CONFORMAL MAPPINGS and how to use them to solve physical and engineering problems in Potential Theory and fluid flow in the plane. On Friday, we will look at Laplace transforms from the complex point of view and derive a formula for the inverse inverse Laplace transform using a Bromwich contour integral. |
09/02 to 13/02 | Sections 25.5 |
25.9
Exercises: 25.12, 25.14, 25.15 |
TEST #1 will be
held on Tuesday, February 10th from 19:05 to 19:55 in
BSB 137, 138. The test will cover the material that
was done in class up to and including the lecture on
Tuesday, February 3rd. On Tuesday I will give a review and finish the discussion about Bromwich contour integrals. On Thursday and Friday, we will have a (quick) look at Stokes' equation, Airy integrals and WKB methods. I will give an application of Airy integrals to quantum tunneling. Next week is Mid-term Recess |
23/02 to 27/02 | Sections 25.6, 25.7, 25.8 | 25.9 Exercises: 25.16, 25.17. 25.18, 25.19, 25.20, 25.21, 25.22, 25.23 |
On Tuesday, I will wrap
up complex analysis by giving a brief overview of general
steepest descent and stationary phase
methods to approximate integrals. These
are essential techniques in Quantum Mechanics. Assignment #3 is due this week on Thursday in class. We will begin Probabilty on Thursday! I will define sample space, events, the axioms of probability and simple consequences thereof, such as the exclusion-inclusion principle. The basic combinatorial counting techniques will be explained, including the multinomial formula and some simple applications. |
02/03 to 06/03 | Sections 30.1, 30.2, 30.3, 30.4,
30.5, 30.6 |
30.16
Exercises: 30.3,
30.5, 30.6, 30.9, 30.10, 30.12, 30.14, 30.15,
30.16 |
On Tuesday, I will define the key
concepts of conditional probability, independence
and Bayes'
Formula. The rest of the week is devoted to RANDOM VARIABLES and their generating functions, in particular the characteristic function which is just our old friend, the Fourier Transform. I will define the fundamental notion of a random variable, its expected value (or mean) and its Variance (or Volatility (sic)) |
09/03 to 13/03 | Sections 30.7, 30.8,
30.9, 30.10, 30.11 |
30.16
Exercises:
30.18, 30.20, 30.21, 30.24, 30.25, 30.26, 30.30, 30.32 |
This week I will introduce the most important
probability distributions that are commonly used
together with their characteristic functions: discrete: Bernoulli, binomial, negative binomial, multinomial, hypergeometric, Poisson continuous: Multivariate Normal (Gaussian), exponential, Gamma, Student t, chi-square, Cauchy etc. I will also define the key notion of joint distributions, independence of random variables, marginal distributions, covariance, correlation and prove some basic formulas, such as the expectation and variance of sums of random variables. |
16/03 to 22/03 | Sections 30.12, 30.13, 30.14, 30.15 |
30.16 Exercises: 30.36, 30.37, 30.39, 30.40 |
Assignment #4
is due
this week in class on Tuesday. On Tuesday I will prove the Central Limit Theorem On Thursday I will explain Chebyschev's inequality. Jensen's inequality and do some applications of the CLT. On Friday, I will prove the convolution formula for the pdf of the sum of two random variables and show that the sum of independent Gaussians is a Gaussian, that the sum of independent Gamma's is again a Gamma and that the sum of independent Poisson is again Poisson. |
23/03 to 27/03 | Sections 31.1, 31.2, 31.3, 31.4, 31.5 |
31.8 Exercises: 31.4, 31.5, 31.6, 31.7, 31.8, 31.9, 31.10 |
TEST #2 will
be held tentatively on Tuesday, March 24th from 19:00
to 20:00. The test will cover the material
that was done in class up to and including the lecture
on Tuesday, March 18th. This week, I will begin Statistics. I will explain basic estimators in parametric statistics, their consistency, bias and efficiency and Confidence Intervals. We will also study the maximum likelihood estimator and more about the multivariate normal distribution and its important descendants: chi-squared, student-t and the F-distribution, which play a role in Statistics. We will discuss the Method of Least Squares and Regression and what is known as Hypothesis Testing. |
30/03 to 03/04 |
Sections 31.6, 31.7 Extra material |
31.8 Exercises: 31.14, 31.15, 31.17, 31.20 |
Assignment #5
is due
this week in class on Tuesday. This week we will also discuss some goodness of fit tests including the chi-squared test (which is also useful for testing independence) and the non-parametric Kolmogorv-Smirnov test. |
06/04 to 08/04 | REVIEW |
||