| # | Date | Read | Topic |
| 1. | Tuesday, September 4 | First meeting. Introduction. Complex numbers. Arithmetic operations. Polar representation. de Moivre's formula. Application to sequences defined by recurrent relations. | |
| Thursday, September 6 | No lecture. Will be rescheduled. | ||
| 2. | Tuesday, September 11 | HW 1 given. Sequences and series. Power series. Radius of convergence. etc. Elementary functions: exponent, logarithm. | |
| 3. | Wednesday, September 12 | Elementary functions: power, trigonometric functions. Geometrical properties of elementary functions: fractional function. Stereographic projection. Square root. Riemann surface. | |
| 4. | Tuesday, September 18 | Geometrical properties of elementary functions: exponent, trigonometric functions. Analytic functions. Complex derivative. Cauchy-Riemann equations. Cauchy-Riemann equations in polar form. | |
| 5. | Thursday, September 20 | HW 2 given. 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 25 | Cauchy's integral formula. Cauchy's formulas for derivatives. Mean value theorem. Maximum modulus theorem. Cauchy's inequalities and Liouville's theorem. | |
| 7. | Thursday, September 27 | Taylor series. Laurent series. Isolated singularities of single-valued functions. Removable singularity, poles. | |
| 8. | Tuesday, October 2 | HW 3 given. Removable singularity, poles, and essential singularities. Singularities at infinity. Zeros and poles. Analytic continuation. | |
| 9. | Thursday, October 4 | Identity theorem. Singularity. Monodromy theorem. Contour integration and residue calculus. Residue theorem. Residue at infinity. Examples of integrals. | |
| 10. | Tuesday, October 9 | Logarithm trick. Jordan's Lemma. Examples of integrals. | |
| 11. | Thursday, October 11 | Examples of integrals. Infinite series and infinite products. | |
| 12. | Tuesday, October 16 | Mittag-Leffler's theorem. Examples of series and products. Conformal mapping. Boundary-value problems. Dirichlet and von Neumann boundary-value problems. | |
| 13. | Thursday, October 18 | Take Home Exam is given. Conjugate harmonic functions. Two-dimensional potential problems: electrostatics and magnetostatics, heat flow. Potential, stream function, and complex potential. Two-dimensional potential problems: hydrodynamics of an ideal incompressible liquid. Example: streamlining a dam. | |
| 14. | Tuesday, October 23 | HW 4 given. Invariance of Laplace equation under analytic transformations. Generalized boundary-value problems (prescribed singularities). Examples of boundary-value problems. Method of images. | |
| 15. | Thursday, October 25 | Streamlining a cylinder. Conformal transformations. Mapping of domains. Riemann's theorem. | |
| 16. | Tuesday, October 30 | Dirichlet problem on a circle. Poisson kernel. Examples of conformal mappings. Bilinear transformation. | |
| 17. | Thursday, November 1 | Examples of conformal mappings. Square root. Exponent. Joukowsky transformation. Streamlining a plane wing. Trigonometric functions. | |
| 18. | Tuesday, November 6 | 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. Asymptotic methods. Example of asymptotic expansion (exponential integral). | |
| 19. | Thursday, November 8 | HW 5 given. Asymptotic expansions from the integration by parts. The failure of the integration by parts Laplace's method (examples). | |
| 20. | Tuesday, November 13 | Watson lemma. Fourier-type integrals. Laplace's method. Intermediate asymptotics. Method of steepest descents (saddle-point method). | |
| 21. | Thursday, November 15 | Method of steepest descents (saddle-point method). Method of stationary phase. | |
| 22. | Tuesday, November 20 | HW 6 given. Asymptotic evaluation of sums. Approximation by Riemann integral. Euler-Maclaurin summation formula. Example: Stirling's formula. Special functions. Introduction: special functions and symmetries. Gamma function. Euler's formula. | |
| Thursday, November 22 | NO CLASSES. Thanksgiving. | ||
| 23. | Tuesday, November 27 | Integral representations by Euler and Hankel. Infinite product representations by Gauss and Weierstrass. Properties of Gamma function. Asymptotic expansion of Gamma function. | |
| 24. | Thursday, November 29 | Digamma function. Beta function. Pochhammer's integral. The area of d-dimensional sphere. Bessel's equation. Bessel functions. Series expansion of Bessel functions. | |
| 25. | Tuesday, December 4 | HW 7 given. Last year's final is given (not for credit). 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. Elementary properties: integral representations. | |
| 26. | Thursday, December 6 | Asymptotic behavior of Bessel functions at large argument and large order. Example of applications: eigenmodes of the circular drum. Generalized hypergeometric functions. Basics of topology. Topology and topological space. Metric topology. | |
| 27. | Tuesday, December 11 | Homeomorphism. Topological properties. 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. | |
| 28. | Thursday, December 13 | Higher homotopy groups. 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. | |
| 29. | Monday, December 17 | Additional lecture. 10am-12pm, B-131. Higher homotopy groups. 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. | Thursday, December 20 | Final exam. 9:30am-12:00pm, B-131 (Physics Building) |
