Physics 503: Methods of Mathematical Physics

Syllabus (tentative)

# Date Read Topic
1. Thursday, September 7 First meeting. Introduction. Complex numbers. Arithmetic operations. Polar representation. de Moivre's formula. Application to sequences defined by recurrent relations.
2. Tuesday, September 12 Sequences and series. Power series. Radius of convergence. etc. Elementary functions: exponent, logarithm, power.
3. Thursday, September 14 HW 1 given. Elementary functions: trigonometric functions. Geometrical properties of elementary functions: fractional function. Stereographic projection.
4. Thursday, September 14 Riemann surface. Analytic functions. Complex derivative. Cauchy-Riemann equations. Cauchy-Riemann equations in polar form.
5. Tuesday, September 19 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 26 HW 2 given. Cauchy's integral formula. Cauchy's formulas for derivatives. Mean value theorem. Maximum modulus theorem.
7. Tuesday, September 26 Cauchy's inequalities and Liouville's theorem. Taylor series. Laurent series.
8. Thursday, September 28 Isolated singularities of single-valued functions. Removable singularity, poles, and essential singularities. Singularities at infinity.
9. Tuesday, October 3 HW 3 given. Zeros and poles. Analytic continuation. Identity theorem. Singularity. Monodromy theorem. Contour integration and residue calculus. Residue theorem. Examples of integrals.
10. Tuesday, October 3 Residue at infinity. Examples of integrals. Logarithm trick. Jordan's Lemma.
11. Thursday, October 5 Jordan's Lemma. Examples of integrals. Infinite series and infinite products.
12. Tuesday, October 10 Infinite series and infinite products. Mittag-Leffler's theorem. Examples of series and products.
13. Tuesday, October 10 HW 4 given. Conformal mapping. Boundary-value problems. Dirichlet and von Neumann boundary-value problems. Conjugate harmonic functions. Two-dimensional potential problems: electrostatics and magnetostatics, heat flow. Potential, stream function, and complex potential.
14. Tuesday, November 7 Two-dimensional potential problems: hydrodynamics of an ideal incompressible liquid. Example: streamlining a dam. Invariance of Laplace equation under analytic transformations. Generalized boundary-value problems (prescribed singularities).
15. Tuesday, November 7 Examples of boundary-value problems. Method of images. Streamlining a cylinder. Conformal transformations.
16. Thursday, November 9 Mapping of domains. Riemann's theorem. Dirichlet problem on a circle. Poisson kernel. Examples of conformal mappings. Bilinear transformation.
17. Tuesday, November 14 Examples of conformal mappings. Square root. Exponent. Joukowsky transformation. Streamlining a plane wing. Trigonometric functions. Schwartz-Christoffel transformation: mapping the interior of a polygon into the upper half-plane.
18. Tuesday, November 14 Schwartz-Christoffel transformation: mapping the exterior of a polygon into the exterior of a unit circle. Causality and Kramers-Kronig (dispersion) relations. Asymptotic methods. Example of asymptotic expansion (exponential integral).
19. Thursday, November 16 Take Home Exam is given. Integration by parts. Laplace's method (examples). Watson lemma.
20. Tuesday, November 21 HW 5 given. Fourier-type integrals. Laplace's method. Intermediate asymptotics. Method of steepest descents (saddle-point method).
21. Tuesday, November 21 Method of steepest descents (saddle-point method). Method of stationary phase. Asymptotic evaluation of sums. Approximation by Riemann integral. Euler-Maclaurin summation formula. Example: Stirling's formula. Special functions. Introduction: special functions and symmetries.
Thursday, November 23 NO CLASSES. Thanksgiving.
Tuesday, November 28 NO CLASSES. Will be rescheduled.
22. Thursday, November 30 HW 6 given. Gamma function. Integral representations by Euler and Hankel. Infinite product representations by Gauss and Weierstrass. Properties of Gamma function. Asymptotic expansion of Gamma function.
23. Tuesday, December 5 Digamma function. Beta function. Pochhammer's integral. The area of d-dimensional sphere. Bessel's equation. Bessel functions. Series expansion of Bessel functions.
24. Tuesday, December 5 Bessel functions of first and second kinds (J,Y). Bessel functions of the third kind (H). Modified Bessel functions (I,K). Elementary properties: recurrence relations, generating function.
25. Thursday, December 7 HW 7 given. Last year's final is given (not for credit). Elementary properties: integral representations. Asymptotic behavior of Bessel functions at large argument and large order. Example of applications: eigenmodes of the circular drum. Generalized hypergeometric functions.
26. Tuesday, December 12 Basics of topology. Topology and topological space. Homeomorphism. Topological properties. Examples of topological spaces.
27. Tuesday, December 12 Examples of topological spaces. Homotopy theory. Zeroth homotopy group. Topological (homotopy) classes. Fundamental group. Homotopic equivalence. Examples of fundamental groups for topological spaces. Examples of fundamental groups for topological spaces. Free homotopy classes of loops vs. fundamental group. Higher homotopy groups.
28. Thursday, December 14 Long (if desired) lecture. Examples of higher homotopy groups for topological spaces. Topological theory of defects in ordered media. Order parameter space. Point defects, line defects, and domain walls. Homotopy classification of topological defects. Textures. Examples of topological defects and textures of ordered media.
Final. Monday, December 18 Final exam. 10am - 12:30pm in B-131.