Meeting no. | ObjectivesAfter 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(z _{0}) in the neighborhood of the zero z_{0} 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 | |

2^{nd} 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 | |

3 ^{rd} 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 | |

4 ^{th} 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 Î¸ and Ï† as a sum over spherical harmonics § Express 1/ │ x_{1}-x_{2}│ 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 | |

5 ^{th} 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 | |

6^{th} 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.

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