Syllabus for Math 3D03 (Mathematical Physics II),
Term2, 2013/14

The following is a tentative syllabus for the course. This page will be updated regularly.
The chapters and sections refer to the text book "Mathematical Methods for Physics and Engineering" by
K.F. Riley, M.P. Hobson & S.J. Bence.

Week Sections in Text Suggested Homework Comments
06/01 to 10/01
Review of Chapter 3 and 4.
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, 3.26, 3.28
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 satisfy. 
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
Sections 24.13, 25.4

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 integration techniques.
 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

On Tuesday we will look at Laplace transforms from the complex point of view and derive a formula for the inverse transform using a Bromwich contour integral
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, 25.8
Selected sections from Chapters  22 and 29
25.9 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, February 6th.
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

Assignment #3   is 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, 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,  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, 30.40

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

Assignment #4  is 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
31.5, 31.6, 31.7

31.8 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

Extra material

Assignment #5  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