|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, 24.8, 24.9
|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 very
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 satisfy.
On Tuesday, we will integrate complex functions, or better, one forms f(z)dz along 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 Thursday, I will derive the CAUCHY INTEGRAL FORMULA, which is the key ingredient in complex analysis
|13/01 to 17/01
||Sections 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
|This week we will discuss the 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: 24.11, 24.12, 24.13, 24.14, 24.15, 24.16, 24.17, 24.18, 24.19, 24.20, 24.21, 24.22
25.9 Exercises: 25.9, 25.10, 25.11
due this week on Tuesday at the beginning of
the lecture period.
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 integration techniques.
|27/01 to 31/01
||Sections 25.5, 25.3||
25.9 Exercises: 25.3, 25.4, 25.5, 25.6, 25.7, 25.8, 25.12, 25.13, 25.14, 25.15
Monday we will look
at Laplace transforms from the complex
point of view and derive a formula for the
inverse transform using a Bromwich
On Tuesday and Thursday 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 24.7, 25.1, 25.2||
25.9 Exercises: 25.2, 25.3, 25.4, 25.5, 25.16, 25.17, 25.18
| Assignment #2
is due on Tuesday at the beginning of the
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.
|10/02 to 14/02||Sections 25.6, 25.7,
Exercises: 25.19, 25.20, 25.21, 25.22, 25.23
||TEST #1 will be
held on Tuesday, February 11th from 19:00 to 20:00 in
T13. 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.
Next week is Mid-term Recess
|24/02 to 28/02||Sections 30.1, 30.2, 30.3, 30.4, 30.5, 30.6, 30.7||
30.16 Exercises: 30.3, 30.5, 30.6, 30.9, 30.10, 30.12, 30.14, 30.15
|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
On Thursday I will define the key concepts of conditional probability, independence and Bayes' Formula.
|03/03 to 07/03||Sections 30.8, 30.9, 30.10, 30.11, 30.12|| 30.16
Exercises: 30.18, 30.20, 30.21, 30.24, 30.25,
30.26, 30.30, 30.32
||Assignment #3 is due this week on
Tuesday in class.
This 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 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, Student t, chi-square, Cauchy etc.
I will 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. On Thursday I will prove the Central Limit Theorem and Chebyschev's inequality.
|10/03 to 14/03||Sections 30.13, 30.14, 30.15, 31.1, 31.2||30.16
Exercises: 30.36, 30.37, 30.39, 30.40
||On Monday I will
inequality and do some applications of the
On Tuesday, 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 and some applications of the CLT. including the convolution f On Thursday, I will begin Chapter 31: Statistics.
|17/03 to 21/03||Sections 31.3, 31.4, 31.5, 31.6
31.8 Exercises: 31.4, 31.5, 31.6, 31.7, 31.8, 31.9, 31.10
| Assignment #4
this week in class on Tuesday.
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.
|24/03 to 28/03||
31.8 Exercises: 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 Thursday, March 20th.
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. On Thursday, I will give a short introduction to Stochastic Processes and tie up some extra material about Statistics (for example, about the Cramer-Rao lower bound and the Fisher Information Matrix)
31/03 to 04/04
29.12 Exercises: 29.2, 29.3, 29.7
| Assignment #5
this week in class on Tuesday.
On Monday and Tuesday, I will give a very brief introduction to groups and their representations (understanding symmetries is fundamental for both physics and mathematics)
|07/04 to 08/04|| REVIEW
||Review of complex analysis on Monday and review of
Probability/Statistics on Tuesday.