About Me

When not at work with students, I spend my time in my room either reading, calculating something using pen and paper, or using a computer. I read almost anything: from the pornographic to the profound, although my main interests are mathematics and physics. "When I get a little money I buy books; and if any is left I buy food and clothes." -Erasmus

Sunday, June 26, 2011

reading guides and complex methods

I'm teaching complex (variable) methods this semester, and I'm using Arfken's book to teach it. It's a long list of topics; the detailed syllabus is shown below:


Meeting no.
Objectives
After the discussion and lined up activities, you should be able to:

Topics
1
(6-15)

§                 Explain what is expected of you to get good marks in this class
§                 Explain the expected role of your teacher
§                 Explain the expected role of your book
§                 Explain the expected role of your lecture classes
§                 List the materials you will need for this course
§                 Explain why the rules of coherence are needed.

Orientation

 Read:  Rules of Coherence

2
(6-17)
§                 Perform algebraic operations using complex numbers
§                 Use the geometric series to evaluate complex valued sums
§                 Use the natural logarithm function to calculate inverse trigonometric functions
§                 Use the natural logarithm function to calculate complex powers

Complex Algebra

Complex-valued Series

Complex Powers

Read: Sec 6.1
Exercises: 6.1.7;  6.1.8;  6.1.9;  6.1.10;  6.1.16, 6.1.17; 6.1.22


DIAGNOSTIC EXAMINATION
12June 18, 2011 (Sat) 1pm


Complex analysis i

3
(6-22)

§                 Use the definition of the derivative to evaluate the derivative of polynomials
§                 Use the definition of the derivative to obtain the Cauchy-Riemann conditions
§                 Use the Cauchy-Riemann conditions and continuity of partial derivatives to test a complex function for analyticity.
§                 Use the Cauchy-Riemann conditions to show that the real and imaginary parts of an analytic function must satisfy the Laplace equation

Cauchy-Riemann conditions

Laplace equation

Analytic functions


Read: Sec 6.2
Exercises: 6.2.1; 6.2.4; 6.2.5

4
(6-24)

§                 Define the Riemann contour integral on a complex contour and compare with the real Riemann integral
§                 Use the definition of the Riemann integral to prove the Darboux inequality
§                 Use the definition of the contour integral to show that, in general, the contour integral is path-dependent.
§                 Use the definition of the contour integral to calculate the contour integral of a given function along a given path.
§                 Prove the Cauchy-Integral Theorem using Stokes’ theorem

Contour Integral

Darboux inequality

Path-dependence

Cauchy Integral Formula

Read:  Sec 6.3  
Exercises: 6.3.1; 6.1.2; 6.3.3

5
(6-29)

§                 Prove the Cauchy integral formula by using a deformation of contour and polar coordinates.
§                 Use the Cauchy integral formula to obtain an expression for the nth derivative of an analytic function.
§                 Use the Cauchy integral formula to relate Rodrigues formula representations of special functions to contour integral representations.

Cauchy Integral formula

Nth derivative of an analytic function

Contour integral representations


Read: Sec 6.4
Exercises:  6.4.1; 6.4.5; 6.4.6; 6.4.8

6
(7-1)

§                 Derive the Taylor series of given functions and give the domain of validity
§                 Derive the Laurent series of given functions about a given point on the complex plane and, using knowledge of singularities, give the domains of validity

Taylor series
Laurent series
Singularities


Read Section 6.5
Exercises 6.5.1; 6.5.2; 6.5.3; 6.5.7; 6.5.8; 6.5.10; 6.5.11

7
(7-6)
§                 Obtain a formula for the coefficient a-1 in terms of derivatives
§                 Given a curve in the z-plane, and a mapping w(z), obtain the corresponding curve on the complex w-plane
§                 Given a simply connected-region A, and a given mapping w(z), obtain its image on the w-plane
Residue formula

Mapping

Read Section 6.6, 6.7
Exercises: 6.6.1, 6.6.2, 6.7.6


First long exam
july 16, 2011 (sat) 1pm

COMPLEX ANALYSIS II
8-9
(7-8 to 7-13)

§                 Identify the poles and other singularities of a given meromorphic function, and evaluate the residues at these singularities (if any)
§                 Use the residue theorem to evaluate a closed simply connected contour integral
§                 Use the residue theorem and Darboux’s inequality to calculate Fourier integrals
§                 Use the residue theorem to calculate other integrals.

Singularities and residues

Residue at infinity

Contour integration

Read Sec 7.1 (exclude Pole and product expansions, pp 461-462)
Exercises 7.1.1, 7.1.3, 7.1.6, 7.1.8, 7.1.10, 7.1.11, 7.1.14, 7.1.15, 7.1.17, 7.1.18, 7.1.21, 7.1.24

10
(7-15)

§                 Given a series for w(z) in powers of z, obtain the series for the inverse function of z(w) in powers of w.
§                 Give the conditions for expanding an analytic function as a pole expansion
§                 Obtain the pole expansion of selected functions
§                 Obtain the product expansions of sin(z) and cos(z)


Inversion of Series

Pole Expansions

Product Expansions

Summation of Series via Contour Integrals



Read Whittaker and Watson, pp 134 to 139, Arfken  pp 461-462, Morse and Feshbach pp 411-413,pp 413-414
Exercises Whittaker and Watson Section 7.4 Examples 4, 6 and Section 7.5 Example 1
Exercises for inversion of series will be handed out in class




11 to 12
(7-20 to 7-22)
§                 Find the zeroes of the derivative of an analytic function f(z), and draw contour plots of the real and imaginary parts of f(z)-f(z0) in the neighborhood of the zero z0 obtained.
§                 Use the method of steepest descent to approximate a class of integrals with parameter s, for large values of s.
§                 Obtain the asymptotic series using the method of steepest descent

Method of Steepest Descent

Read Sec 7.3, Morse and Feshbach pp 434-443
Exercises 7.3.1, 7.3.2, 7.3.3, exercises to be handed out


2nd Long Exam
July 30, 2011 (Sat) 1pm

gAMMA, BETA AND FOURIER SERIES

13
(7-27)

§                 Show the equivalence of  the three definitions (Euler integral analytically continued; Weierstrass product, Euler product) of the Gamma function
§                 Use the Gamma function to evaluate negative factorials
§                 Use the Gamma function to obtain the Binomial series valid for non-integral powers, and give the domain of validity
§                 Derive the recursion relation of the gamma function
§                 Find the poles of the gamma function and evaluate the residues of the gamma function
§                 Prove the Gauss-Multiplication theorem and the Legendre duplication formula
§                 Evaluate Gaussian integrals

Gamma Function

Binomial series

Gaussian integrals

Multiplication Formula




Read Sec 8.1, and Whittaker and Watson pp 244-246
Exercises  8.1.1, 8.1.4, 8.1.5, 8.1.9, 8.1.14,  8.1.18, 8.1.24

14
(7-29)

§                 Derive Stirling’s series
§                 Use Stirling’s approximation to evaluate large factorials.
§                 Use the Beta function and the chain rule to evaluate a selected class of integrals
Stirling’s Approximation

Beta Function

Read sec 8.3 and 8.4, Whittaker and Watson pp 251 to 253
Exercises 8.3.1, 8.3.6, 8.3.8, 8.3.9,  8.4.2, 8.4.17, 8.4.18

15-16
(8-3 to 8-5)

§                 Prove the orthogonality of a given set of sines and cosines on a suitable interval
§                 Use orthogonality and completeness of a basis of sines and cosines to obtain the Fourier series of a function within an interval
§                 Use Fourier series to solve the wave equation of a vibrating string
Fourier Series

Orthogonality



Read Sec 14.1 to 14.4
Exercises 14.1.5, 14.1.9, 14.2.1, 14.2.3, 14.3.2, 14.3.12, 14.3.14, 14.4.2, 14.4.10


3rd Long Exam
12August 13, 2011 (Sat) 1pm

INTEGRAL TRANSFORMS
17
(8-10)


§                 Expand given functions defined over the whole real line as a Fourier integral
§                 Obtain the Fourier transform of a given function
§                 Given the function in (Fourier) k-space, use the inverse Fourier transform to obtain the function in coordinate space
§                 Relate Mellin transforms to Fourier transforms
§                 Obtain the representation of the Dirac delta in terms of Fourier integrals

Fourier Transforms



Read Section 15.1 to 15.3
Exercises 15.1.3, 15.3.2, 15.3.4, 15.3.9, 15.3.17,

18
(8-12)
§                 Obtain the Fourier transform of derivatives
§                 Convert a linear differential equation in coordinate space into the corresponding integral equation in k-space
§                 Solve the diffusion equation using Fourier transforms

Fourier transform of derivatives

Read Sec 15.4
Exercises 15.4.1, 15.4.3, 15.4.4, 15.4.5

19
(8-17)

§                   Use the convolution theorem to evaluate some integrals
§                   Obtain the momentum space representation of a wavefunction


Convolution Theorem
Parseval’s Relation
Momentum Space

Read Sections 15. 5 to 15. 6
Exercises 15.5.3, 15.5.5, 15.5.6, 15.5.8, 15.6.3, 15.6.8, 15.6.12


20
(8-24)

§                   Calculate the Laplace transform of some elementary functions
§                   Use tables of Laplace transforms and the linearity of Laplace transforms to evaluate inverse Laplace transforms
§                   Use partial fractions to evaluate inverse Laplace transforms

Laplace Transform

Partial fractions




Read Section15.8
Do Exercises 15.8.3, 15.8.4, 15.8.5, 15.8.9

21
(8-26)
§                   Evaluate the Laplace transform of derivatives
§                   Convert linear differential equations with constant coefficients to algebraic systems
§                   Solve linear ordinary differential equations with constant coefficients using Laplace transforms


Laplace Transform of Derivatives

Convolution Theorem

Read Section 15.9 to 15.11
Exercises 15.9.2, 15.9.3, 15.11.2, 15.11.3

22
(8-31)
§                   Evaluate inverse Laplace transform using Bromwich integrals
§                   Convert linear differential equations with constant coefficients to algebraic systems
Bromwich integral

Read Section 15.12
Exercises 15.12.1, 15.12.2, 15.12.3, 15.12.4


4th Long Exam
September 3, 2011 (Sat)

SPECIAL FUNCTIONS I – BESSEL, LEGENDRE FUNCTIONS
23 to 24
(9-2 to 9-7)

§          Identify the singularities of linear second order differential equations
§          Use the power-series method to obtain a solution of linear second order differential equations
§           Use Wronskians to obtain a linearly independent second solutions if a solution is known



Power-Series Solutions



Read: Section 9.4 to 9.6
Ex: 9.4.1,9.4.2, 9.4.3, 9.5.5, 9.5.6, 9.5.10, 9.5.11, 9.6.18, 9.6.19, 9.6.25

25
(9-9)
§          Use generating functions to obtain the recursion relations satisfied by Bessel Functions
§          Use power-series methods to solve Bessel’s differential equation for both integral and non-integral powers
§          Use the generalized Green’s theorem to verify the orthogonality properties of a set of Bessel functions
§          Use the completeness of a set of Bessel functions to expand a given function in the interval 0≤ x ≤ a
Bessel Functions of the First Kind

Generating Function



Read: Sec 11.1 to 11.2
Ex: 11.1.1, 11.1.3, 11.1.10, 11.1.16, 11.1.18, 11.2.2, 11.2.3, 11.2.6

26
(9-14)

§          Use the definition of Neumann functions and verify that it is a second linearly independent solution of Bessel’s equation
§          Use the definition of Hankel functions to derive its properties
§          Use the asymptotic formulae for Bessel functions to approximately evaluate Bessel functions for large values of its argument
Neumann and Hankel functions

Asymptotic formulae for Bessel functions


Read: Sec 11.3 to 11.4
Ex: 11.3.2,11.3.6, 11.4.7

27
(9-16)

§                 Use generating functions to obtain recursion relations and other properties of Legendre functions
§                 Use the generalized Green’s  theorem to prove orthogonality of Legendre functions
§                 Expand an arbitrary function within the interval -1≤ x ≤  in terms of Bessel functions and give an integral for the expansion coeffiecients
§                 Use Rodrigues formula to derive orthogonality of Legendre polynomials and calculate the normalization constant of Legendre polynomials

Legendre functions

Orthogonality and Completeness

Read: 12.1 to 12.4
Ex:  12.2.2, 12.2.3, 12.2.5, 12.3.2,12.3.6, 12.3.11, 12.4.2

28
(9-21)
§                 Prove orthogonality of associated Legendre functions
§                 Use completeness relations of Spherical Harmonics to express functions depending on θ and φ  as a sum over spherical harmonics
§                 Express 1/ │x1-x2│ in terms of spherical harmonics
Associated Legendre functions

Spherical Harmonics

Addition Theorem

Read sec 12.5 to 12.6, 12.8
Exercises 12.5.1, 12.5.11,12.6.4, 12.6.5,12.8.3, 12.8.8

 

5th Long Exam
September 24, 2011 (Sat)

SPECIAL FUNCTIONS II—HERMITE, LAGUERRE, HYPERGEOMETRIC FUNCTIONS
29
(9-23)
§                 Prove orthogonality of Hermite functions
§                 Solve Hermite’s differential equation via power series
§                 Use completeness relations to express functions in the interval -∞ ≤ x  ≤∞ as a sum of Hermite polynomials
§                 Use generating function to obtain the Rodrigues formula for Hermite polynomials
§                 Use Rodrigues formula to obtain Hermite polynomials
Hermite functions

Completeness and Orthogonality

Rodrigues and Integral Representations

Read: Sec 13.1
Exercises 13.1.2, 13.1.14, 13.1.12, 13.1.13

30
(9-28)
§                 Prove orthogonality of Laguerre functions
§                 Use completeness relations to express functions in the interval 0≤ x  ≤∞ as a sum of Laguerre polynomials
§                 Use generating function to obtain the Rodrigues formula for Laguerre polynomials
§                  Use Rodrigues formula to obtain Laguerre polynomials
Laguerre polynomials

Associated Laguerre functions

Read: Sec 13.2
Exercises 13.2.1, 13.2.3, 13.2.6, 13.2.7




31
(9-30)
§                 Prove orthogonality of Chebyshev functions
§                 Use completeness relations to express functions in the interval -1≤ x  ≤1 as a sum of Chebyshev polynomials
§                 Use generating function to obtain the Rodrigues formula for Chebyshev polynomials
§                 Use Rodrigues formula to obtain Chebyshev polynomials

Chebyshev Polynomials

Read Sec 13.3
Exercises 13.3.1, 13.3.3, 13.3.5

32
(10-5)
§                 Reduce any linear second order differential equation with three regular singularities into the hypergeometric equation
§                 Express some orthogonal special functions in terms of hypergeometric fuctions

Hypergeometric Functions

Read Sec 13.4
Exercises 13.4.6, 13.4.7, 13.4.8

33
(10-7)
§                 Reduce any linear second order differential equation with one regular singularity and one irregular singularity into the confluent hypergeometric equation
§                 Express some special functions in terms of confluent hypergeometric functions

Confluent Hypergeometric Functions

Read Sec 13.5
Exercises 13.5.4,  13.5.6, 13.5.11, 13.5.13, 13.5.14



6th Long Exam
October 15, 2011 (Sat) 1pm



The pace is very fast. When I first learned these topics, it took me more than two years of work, mainly because I had no one to look over my work. I used Churchill's book, Complex Variables with Applications, but  it's not enough to give justice to the course description. And I need to follow the course description because of university rules.

If asked, I would be the first to agree that it's an unreasonable amount of material, especially for a course that meets three hours a week. Cambridge University, for example offers a Methods of Mathematical Physics Course. One set of notes I found includes all the complex variable material, and removes most of the special functions. The only special functions that do show up are the hypergeometric and confluent hypergeometric functions, and Gamma and Beta. No discussion of Fourier series or Fourier integrals!

I am, however, stuck with the course description. I think that there ought to be changes made, but it has to go up to the university council. The only reasonable way of covering this material is to give them lots of work to do at home, and make extensive use of consultation hours.

I find it difficult to prepare conventional lecture notes. My main objection to conventional lecture notes is that reading the lecture notes is a more passive activity compared to working with pen in hand to prove the theorems or solve the problems. So instead of lecture notes, I will prepare reading guides.

The reading guides are a series of tasks that one should do while reading the text. One of the objections to Arfken is how easy it is to get lost. The way to avoid it is to divide the section into parts, and as soon as one reads the subparts, one should work on a problem or two in Arfken. I've prepared the reading guide so that when my students actually follow the guide, they will be able to construct a decent set of notes, and at the same time, solve the problems I've listed on the syllabus.

I hope that the reading guide makes it easier to read Arfken's book. My own method of reading Arfken (since I had no guide before) was to attempt solving the problems at the end of the section before reading the section.  But that takes more time, since I could not separate the more important problems from the ones of secondary interest. Even if my students solve all of the problems I've listed, it's still a fraction of what I've actually done on my own.

I hand out the reading guides a week or so before the lecture class, and I expect my students to use the reading guide to prepare for the coming class. This means I will not need to discuss everything; instead, I could concentrate on the more difficult parts.

My students are graduate students with back subjects-- they had their undergraduate courses elsewhere, and are in need of remediation. They had no complex methods courses during their undergraduate days. Since they're graduate students, they have, at most, 9 hours of classwork every week. I hope that all of them are full-time students; the pace we set will be demanding. But if they do the necessary work, they should end the semester with an unfair advantage over their classmates in other physics courses.