| 1. | Tuesday, August 28 |
First meeting.
Introduction. Complex numbers.
Arithmetic operations.
Polar representation. de Moivre's
formula. Application to
sequences defined by recurrent relations. |
| 2. | Thursday, August 30 |
HW 1 given.
Sequences and series. Power
series. Radius of convergence. etc. Elementary functions: exponent,
trigonometric functions, logarithm. |
| 3. | Tuesday, September 4 |
Elementary functions:
power. Geometrical properties of elementary functions: fractional
function, exponent, power. |
| 4. | Thursday, September 6 |
Complex derivative.
Analytic functions.
Cauchy-Riemann equations. |
| 5. | Tuesday, September 11 |
Cauchy-Riemann equations
in polar and complex forms. Harmonic functions. |
| 6. | Thursday, September 13 |
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 20 |
Cauchy's formulas for
derivatives. Mean value theorem. Maximum modulus theorem. Cauchy's
inequalities and Liouville's theorem. Taylor series. Laurent series. |
| 8. | Tuesday, September 25 |
HW 3 given.
Laurent series. Isolated
singularities of single-valued functions. Removable singularity, poles,
and essential singularities. Singularities at infinity. Zeros and
poles. |
| 9. | Tuesday, October 2 |
Analytic continuation.
Identity theorem. Singularity. Monodromy theorem. Residue theorem.
Contour integration.
Residue calculus. Examples of integrals. |
| 10. | Thursday, October 4 |
HW 4 given.
Residue calculus. Residue at infinity.
Examples of integrals. |
| 11. | Tuesday, October 9 |
Jordan's Lemma.
Examples of integrals. |
| 12. | Thursday, October 11 |
Examples of integrals. Infinite series and infinite products. |
| 13. | Tuesday, October 16 |
Conformal mapping.
Two-dimensional potential problems: hydrodynamics of an ideal
incompressible liquid, electrostatics and magnetostatics, heat flow.
Potential, stream function,
and complex potential. Conjugate harmonic functions. |
| 14. | Thursday, October 18 |
HW 5 given. 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.
Streamlining a cylinder.
|
| 15. | Tuesday, October 23 |
Conformal transformations. Mapping of domains. Riemann's theorem.
|
| 16. | Thursday, October 25 |
Dirichlet problem on a circle. Poisson kernel. Examples of
conformal mappings. Bilinear transformation. Exponent. Joukowsky
transformation. Streamlining a plane wing.
|
| 17. | Tuesday, October 30 |
Examples of
conformal mappings. Trigonometric functions. Schwartz-Christoffel
transformation: mapping the interior of a polygon into the upper half-plane
and the exterior of a polygon into the exterior of a unit circle.
Causality and Kramers-Kronig (dispersion) relations.
|
| 18. | Thursday, November 1 |
HW 6 given.
Asymptotic methods.
Asymptotic expansion. Integration by parts. Laplace's method. Watson lemma.
|
| 19. | Tuesday, November 6 |
Fourier-type integrals. Method of steepest descents (saddle-point
method).
|
| 20. | Thursday, November 8 |
Method of steepest descents. Method of stationary phase.
Asymptotic evaluation of sums. Approximation by Riemann integral.
Euler-Maclaurin summation formula. Example: Stirling's formula.
|
| 21. | Tuesday, November 13 |
Special functions. Introduction:
special functions and symmetries. Gamma function. Integral representations
by Euler and Hankel. Infinite product representations by Gauss and
Weierstrass. Properties of Gamma function.
|
| 22. | Thursday, November 15 |
HW 7 given.
Asymptotic expansion of Gamma function. Digamma function. Beta function.
Pochhammer's integral. The area of d-dimensional sphere. Bessel's equation.
|
| 23. | Tuesday, November 27 |
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,
integral representations.
|
| 24. | Thursday, November 29 |
HW 8 given. Asymptotic behavior of
Bessel functions at large argument and large order. Example of
applications: eigenmodes of the circular drum. Generalized hypergeometric
functions.
|
| 25. | Tuesday, December 4 |
Basics of topology. Topology and
topological space. Homeomorphism. Topological properties.
Examples of topological spaces.
|
| 26. | Thursday, December 6 |
Examples of topological spaces. Homotopy theory.
Zeroth homotopy group. Topological (homotopy) classes.
Fundamental group. Homotopic equivalence. Examples of fundamental groups
for topological spaces.
|
| 27. | Tuesday, December 11 |
HW 9 given (not for credit).
Examples of fundamental groups for topological spaces. Free homotopy
classes of loops vs. fundamental group. Higher homotopy groups. Examples
of higher homotopy groups for topological spaces.
|
| 28. | Thursday, December 13 |
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.
11:00am - 1:30pm in P-128.
|