| 1. | Thursday, September 4 | First meeting. Introduction. Complex numbers. Arithmetic operations. Polar representation. de Moivre's formula. Application to sequences defined by recurrent relations. |
| 2. | Tuesday, September 9 | Sequences and series. Power series. Radius of convergence. etc. Elementary functions: exponent, logarithm. |
| 3. | Thursday, September 11 | HW 1 given. Elementary functions: trigonometric functions, power. Geometrical properties of elementary functions: fractional function. |
| 4. | Tuesday, September 16 | Stereographic projection. Riemann surface. Analytic functions. Complex derivative. |
| 5. | Thursday, September 18 | Cauchy-Riemann equations. Cauchy-Riemann equations in polar and complex forms. Harmonic functions. |
| 6. | Tuesday, September 23 | HW 2 given. Integration in complex plane. Independence of integrals on contour in singly-connected and multiply-connected domains of analyticity. Cauchy's integral formula. |
| 7. | Thursday, September 25 | Cauchy's formulas for derivatives. Mean value theorem. Maximum modulus theorem. Cauchy's inequalities and Liouville's theorem. |
| 8. | Tuesday, September 30 | Taylor series. Laurent series. Isolated singularities of single-valued functions. Removable singularity, poles, and essential singularities. |
| 9. | Thursday, October 2 | HW 3 given. Singularities at infinity. Zeros and poles. Analytic continuation. Identity theorem. Singularity. Monodromy theorem. |
| 10. | Tuesday, October 7 | Contour integration and residue calculus. Residue theorem. Residue at infinity. Examples of integrals. |
| 11. | Thursday, October 9 | Examples of integrals. Logarithm trick. Jordan's Lemma. |
| 12. | Tuesday, October 14 | Examples of integrals. Infinite series and infinite products. |
| 13. | Thursday, October 16 | 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, October 21 | 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. | Thursday, October 23 | Generalized boundary-value problems (prescribed singularities). Examples of boundary-value problems. Method of images. Streamlining a cylinder. |
| 16. | Tuesday, October 28 | Conformal transformations. Mapping of domains. Riemann's theorem. Dirichlet problem on a circle. Poisson kernel. |
| 17. | Thursday, October 30 | Examples of conformal mappings. Bilinear transformation. Square root. Exponent. Joukowsky transformation. Streamlining a plane wing. Trigonometric functions. |
| 18. | Tuesday, November 4 | HW 5 given. Schwartz-Christoffel transformation: mapping the interior of a polygon into the upper half-plane. Schwartz-Christoffel transformation: mapping the exterior of a polygon into the exterior of a unit circle. Causality and Kramers-Kronig (dispersion) relations. |
| 19. | Thursday, November 6 | Take Home Exam is given. Asymptotic methods. Example of asymptotic expansion (exponential integral). Integration by parts. Laplace's method (examples). |
| Tuesday, November 11 | NO CLASS | |
| 20. | Thursday, November 13 | Laplace's method. Watson lemma. Fourier-type integrals. Intermediate asymptotics. |
| 21. | Tuesday, November 18 | HW 6 given. Method of steepest descents (saddle-point method). Method of stationary phase. |
| 22. | Wednesday, November 19 | Asymptotic evaluation of sums. Approximation by Riemann integral. Euler-Maclaurin summation formula. Example: Stirling's formula. Special functions. Introduction: special functions and symmetries. |
| 23. | Thursday, November 20 | Gamma function. Integral representations by Euler and Hankel. Infinite product representations by Gauss and Weierstrass. Properties of Gamma function. |
| 24. | Tuesday, November 25 | Asymptotic expansion of Gamma function. Digamma function. Beta function. Pochhammer's integral. The area of d-dimensional sphere. |
| Thursday, November 27 | No classes. | |
| 25. | Tuesday, December 2 | HW 7 given. Last year's final is given (not for credit). 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). Elementary properties: recurrence relations, generating function. |
| 26. | Thursday, December 4 | 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. |
| 27. | Tuesday, December 9 | Basics of topology. Topology and topological space. Homeomorphism. Topological properties. Examples of topological spaces. |
| 28. | Thursday, December 11 | 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. |
| Final. | Tuesday, December 16 | Final exam. 2:00pm - 4:30pm in P-112. |
| 29. | Wednesday, December 17 | Optional 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. |
