Physics 503: Methods of Mathematical Physics

Syllabus (tentative)

# Date Read Topic
1. Tuesday, September 2 First meeting. Introduction. Complex numbers. Arithmetic operations. Polar representation. de Moivre's formula. Application to sequences defined by recurrent relations.
2. Thursday, September 4 HW 1 given. Sequences and series. Power series. Radius of convergence. etc. Elementary functions: exponent, logarithm, power, trigonometric functions. Principal value of logarithm and power.
3. Tuesday, September 9 Geometrical properties of elementary functions: fractional function. Stereographic projection. Square root. Riemann surface. Geometrical properties of elementary functions: exponent, trigonometric functions.
4. Thursday, September 11 Geometrical properties of elementary functions: trigonometric functions. Analytic functions. Complex derivative. Cauchy-Riemann equations. Cauchy-Riemann equations in polar form.
5. Thursday, September 11 Cauchy-Riemann equations in complex form. Harmonic functions. Integration in complex plane. Independence of integrals on contour in singly-connected and multiply-connected domains of analyticity.
6. Tuesday, September 16 HW 2 given. Cauchy's theorem. Independence of integrals on contour in singly-connected and multiply-connected domains of analyticity. Cauchy's integral formula. Cauchy's formulas for derivatives.
7. Thursday, September 18 Mean value theorem. Maximum modulus theorem. Cauchy's inequalities and Liouville's theorem. Taylor series.
8. Thursday, September 18 Laurent series. Isolated singularities of single-valued functions. Removable singularity, poles, and essential singularities. Singularities at infinity.
9. Tuesday, September 23 Zeros and poles. Analytic continuation. Identity theorem. Singularity. Monodromy theorem. Branching point singularity. Natural boundary. Contour integration and residue calculus. Residue theorem.
10. Thursday, September 25 Residue at infinity. Examples of integrals. Symmetry trick. Logarithm trick.
11. Thursday, September 25 Logarithm trick. Jordan's Lemma. Examples of integrals.
Tuesday, September 30 NO CLASSES. Rosh Hashanah.
12. Thursday, October 2 HW 3 given. Examples of integrals. Contour shift trick. Infinite series and infinite products.
Tuesday, October 7 NO LECTURE (rescheduled).
Thursday, October 9 NO CLASSES. Yom Kippur.
Tuesday, October 14 NO LECTURE (rescheduled).
Thursday, October 16 NO LECTURE (rescheduled).
Tuesday, October 21 NO LECTURE (rescheduled).
Thursday, October 23 NO LECTURE (rescheduled).
13. Tuesday, October 28 HW 4 given. (due Nov. 11) Infinite series and infinite products. Mittag-Leffler's theorem. Examples of series and products.
14. Thursday, October 30 Conformal mapping. Boundary-value problems. Dirichlet and von Neumann boundary-value problems. Conjugate harmonic functions. Two-dimensional potential problems: hydrodynamics of an ideal incompressible liquid, electrostatics and magnetostatics, heat flow. Potential, stream function, and complex potential.
15. Thursday, October 30 Take Home Exam is given. (due Nov. 4) Example: streamlining a dam.Invariance of Laplace equation under analytic transformations. Generalized boundary-value problems (prescribed singularities). Examples of boundary-value problems. Method of images.
16. Tuesday, November 4 Streamlining a cylinder. Conformal transformations. Mapping of domains. Riemann's theorem.
17. Thursday, November 6 Doubly connected domains. Dirichlet problem on a circle. Poisson kernel. Examples of conformal mappings. Bilinear transformation. Examples of conformal mappings. Square root. Exponent.
18. Tuesday, November 11 Joukowsky transformation. Streamlining a plane wing. Blasius' theorem and Kutta condition. Trigonometric functions. Schwartz-Christoffel transformation: mapping the interior of a polygon into the upper half-plane, mapping the exterior of a polygon into the exterior of a unit circle. Causality and Kramers-Kronig (dispersion) relations.
19. Thursday, November 13 HW 5 given. Asymptotic methods. Example of asymptotic expansion (exponential integral). Asymptotic expansions from the integration by parts. The failure of the integration by parts.
20. Thursday, November 13 Laplace's method (examples). Watson lemma.
Tuesday, November 18 NO LECTURE (rescheduled).
21. Thursday, November 20 Fourier-type integrals. Laplace's method. Intermediate asymptotics. Method of steepest descents (saddle-point method).
22. Thursday, November 20 Method of steepest descents (saddle-point method). Method of stationary phase. Asymptotic evaluation of sums. Approximation by Riemann integral. Euler-Maclaurin summation formula.
23. Tuesday, November 25 HW 6 given. Example: Stirling's formula. Special functions. Introduction: special functions and symmetries. Gamma function. Euler's formula. Integral representations by Euler and Hankel. Infinite product representations by Gauss and Weierstrass. Properties of Gamma function.
Thursday, November 27 NO CLASSES. Thanksgiving.
24. Tuesday, December 2 Properties of Gamma function. Asymptotic expansion of Gamma function. Digamma function. Beta function. Pochhammer's integral.
25. Thursday, December 4 Last year's final is given (not for credit). The area of d-dimensional sphere. Bessel's equation. Bessel functions. Series expansion of Bessel functions. Bessel functions of first and second kinds (J,Y). Bessel functions of the third kind (H). Modified Bessel functions (I,K).
26. Tuesday, December 9 HW 7 given. Elementary properties: recurrence relations, generating function. Integral representations. Asymptotic behavior of Bessel functions at large argument and large order. Example of applications: eigenmodes of the circular drum.
27. Thursday, December 11 Generalized hypergeometric functions. Orthogonal polynomials. Example: Quantum oscillator and Hermite polynomials. Orthogonal polynomials corresponding to an arbitrary weight function. Schmidt's process. Kernel. Formula of recurrence.
28. Monday, December 15 Christoffel-Darboux identity. Symmetry. Zeros. Least square property. Differential equation. Rodriguez formula. Families of orthogonal polynomials and their applications. Hermite, Laguerre, Legendre, Gegenbauer, Tchebyshev, Jacobi, etc. Vandermonde determinant and integrals of Slater determinants.
Final. Tuesday, December 23 Final exam. 8:00am-10:30pm, P-128 (Physics Building)