|Week||Sections in Text||Suggested Homework||Comments|
|06/01 to 10/01
||Review of Chapter 3
Sections 24.1, 24.2, 24.3, 24.4, 24.6, 24.8, 24.9, 24.10
|3.8 Exercises: 3.10, 3.12, 3.14, 3.16, 3.18,
4.8 Exercises: 4.16, 4.20, 4.23, 4.33, 4.36
|After a brief
review of some basic facts about complex numbers,
as contained in Chapter 3, and power series as
contained in Chapter 4, we will differentiate complex
functions and derive the Cauchy-Riemann equations that
analytic (or holomorphic) functions have to
On Wednesday, we will integrate complex functions (or better one forms f(z)dz) along closed curves and derive a version of the CAUCHY INTEGRAL THEOREM (the mother of all theorems in complex analysis) with a simple proof based on Green's Theorem. On Friday, we will prove the CAUCHY INTEGRAL FORMULA
|13/01 to 17/01
||Sections 24.3, 24.4, 24.6, 24.10, 24.11, 24.12, 24.13, 24.5||
24.14 Exercises: do all of them
|This week we will discuss ZEROS, POLES, ESSENTIAL SINGULARITIES of complex functions and their TAYLOR and LAURENT series and also the the notion of the winding number and prove the RESIDUE THEOREM, which is the key technique (contour integrals) that is used in complex analysis.|
|20/01 to 24/01
24.14 Exercises: do all of them
|This week we will
evaluate (without using Wolfram alpha!) a number
of important definite integrals such as the Fresnel integral and
sum a few interesting infinite series such as Riemann zeta
function at even numbers, by using contour
Assignment #1 is due this week on Thursday during the lecture period.
|27/01 to 31/01
||Sections 25.5, 25.3, 24.7, 25.1||
25.9 Exercises: 25.3, 25.4, 25.6, 25.7, 25.8, 25.9, 25.10, 25.12, 25.14
Tuesday we will look
at Laplace transforms from the complex
point of view and derive a formula for the
inverse transform using a Bromwich
On Wednesday and Friday we will discuss the mean value property, the maximum modulus principle and the argument principle for analytic functions. The fundamental theorem of algebra will then follow from LIOUVILLE's THEOREM and the Argument Principle can be used to count the number of zeros and poles inside a contour.
|03/02 to 07/02||Sections 25.1, 25.2, 25.4||
25.9 Exercises: 25.16, 25.17, 25.18
|This week will be about
CONFORMAL MAPPINGS and how to use them to solve
physical and engineering problems in Potential Theory and
fluid flow in the plane.
Assignment #2 is due on Thursday during the lecture period.
Friday was SNOWDAY (no classes!)
|10/02 to 14/02||Sections 25.6, 25.7,
Selected sections from Chapters 22 and 29
Exercises: 25.19, 25.20, 25.22, 25.23
22.9 Exercises: 22.4, 22.9, 22.15, 22.16, 22.19, 22.21, 22.24, 22.25, 22.26
29.12 Exercises: 29.2, 29.3, 29.7
|TEST #1 will be
held (tentatively) on Tuesday, February12th from 19:00
to 20:00. The test will cover the material that was
done in class up to and including the lecture on Thursday,
This week, we will have a (quick) look at Stokes' equation, Airy integrals, WKB methods and more generally steepest descent and stationary phase methods to approximate integrals. I will explain what is classically known as "Calculus of Variations" or the "Principle of Least Action" and the Rayleigh-Ritz method for estimating eigenvalues.
Next week is Mid-term Recess
|24/02 to 28/02||Sections 30.1, 30.2, 30.3, 30.4, 30.5|| 30.16
Exercises: 30.3, 30.5, 30.6, 30.9, 30.10, 30.12
due this week on Thursday in class.
On Tuesday I will tie up some loose ends from the first half of the course (to compensate for the snow day ;) and a very very brief introduction to groups and their representations (understanding symmetries is fundamental for both physics and mathematics)
On Wednesday, I will begin Probability and Statistics. 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 to Physics (Bose-Einstein and Fermi-Dirac statistics)
|03/03 to 07/03||Sections 30.6, 30.7, 30.8, 30.9|| 30.16
Exercises: 30.14, 30.15, 30.18, 30.20, 30.21,
30.24, 30.25, 30.26
||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,
value (or mean) and its variance and I
will introduce the following 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, Weibull, Student t, chi-square, Cauchy, Rayleigh and F.
|10/03 to 14/03||Sections 30.10, 30.11, 30.12, 30.13, 30.14, 30.15||30.16
Exercises: 30.30, 30.32, 30.36, 30.37, 30.39,
On Tuesday and Wednesday, I will define the key notion of independence of random variables, marginal distributions, covariance, correlation and prove some basic formulas, such as the expectation and variance of sums of random variables, including the convolution formula for the pdf of the sum of two random variables. We will 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.
On Friday, I will prove the Central Limit Theorem and do some examples.
|17/03 to 21/03||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
due this week in class on Thursday
On Tuesday, I will explain Chebyschev's and Jensen's inequality and do some applications of the CLT.
On Wednesday, 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.
|24/03 to 28/03||
Sections 31.5, 31.6, 31.7
Exercises: 31.10, 31.14, 31.15, 31.17, 31.20
||TEST #2 will be
held tentatively on Tuesday, March 26th from 19:00 to
20:00. The test will cover the material that was
done in class up to and including the lecture on
Wednesday, March 20th.
This week we will discuss the Method of Least Squares, Regression and Hypothesis Testing. I will explain some goodness of fit tests including the chi-squared test (which is also useful for testing independence) and the non-parametric Kolmogorv-Smirnov test.
31/03 to 04/04
is due this week in
class on Thursday.
On Tuesday, I will give a short introduction to Stochastic Processes and Time Series and tie up some loose ends about Statistics (for example, about the Cramer-Rao lower bound and the Fisher Information Matrix)
The rest of the week will be spent reviewing complex analysis.
|07/04 to 08/04|| REVIEW