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
, 24.8, 24.9
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 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
Sections 24.13, 25.4

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
 Assignment #1 is 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

On Monday 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 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 lecture period.

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, 25.8
25.9 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, 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.

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
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 Fermi-Dirac statistics)
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 explain Jensen's inequality and do some applications of the CLT.
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  is due 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

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

Extra material

29.12 Exercises: 29.2, 29.3, 29.7
Assignment #5  is due 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.