| 1. | Thursday, September 5 |
First meeting.
Introduction. Complex numbers.
Arithmetic operations.
Polar representation. de Moivre's
formula. Application to
sequences defined by recurrent relations. |
| 2. | Tuesday, September 10 |
HW 1 given.
Sequences and series. Power
series. Radius of convergence. etc. Elementary functions: exponent,
logarithm. |
| 3. | Thursday, September 12 |
Elementary functions: trigonometric functions, power.
Geometrical properties of elementary functions: fractional
function. |
| 4. | Thursday, September 19 |
Geometrical properties of elementary functions: fractional
function, exponent, power. Stereographic projection.
Analytic functions.
Complex derivative. |
| 5. | Tuesday, September 24 |
HW 2 given.
Cauchy-Riemann equations. Cauchy-Riemann equations
in polar and complex forms. Harmonic functions. |
| 6. | Thursday, September 26 |
Integration in
complex plane. Independence of integrals on contour in singly-connected and
multiply-connected domains of analyticity. Cauchy's integral formula. |
| 7. | Tuesday, October 1 |
Cauchy's formulas for
derivatives. Mean value theorem. Maximum modulus theorem. Cauchy's
inequalities and Liouville's theorem. Taylor series. |
| 8. | Thursday, October 3 |
Laurent series. Isolated
singularities of single-valued functions. Removable singularity, poles,
and essential singularities. Singularities at infinity. |
| 9. | Tuesday, October 8 |
HW 3 given.
Zeros and poles. Analytic continuation.
Identity theorem. Singularity.
Monodromy theorem. Residue theorem.
Contour integration.
Residue calculus. Examples of integrals. |
| 10. | Thursday, October 10 |
Residue calculus. Residue at infinity.
Examples of integrals. |
| 11. | Tuesday, October 15 |
Jordan's Lemma. Examples of integrals.
|
| 12. | Thursday, October 17 |
Examples of integrals.
Infinite series and infinite products. |
| 13. | Tuesday, October 22 |
HW 4 given.
Conformal mapping.
Two-dimensional potential problems. Dirichlet and von Neumann
boundary-value problems. Conjugate harmonic functions.
|
| 14. | Thursday, October 24 |
Two-dimensional potential problems: hydrodynamics of an ideal
incompressible liquid, electrostatics and magnetostatics, heat flow.
Potential, stream function,
and complex potential. Example:
streamlining a dam.
|
| 15. | Tuesday, October 29 |
Invariance of Laplace equation under analytic
transformations. Generalized boundary-value problems (prescribed
singularities). Examples of boundary-value problems. Method of images.
Streamlining a cylinder.
|
| 16. | Thursday, October 31 |
Conformal transformations. Mapping of domains.
Riemann's theorem. Dirichlet problem on a circle.
Poisson kernel. Examples of
conformal mappings. Bilinear transformation.
|
| 17. | Tuesday, November 5 |
Examples of conformal mappings. Exponent.
Joukowsky transformation. Streamlining a plane wing.
Trigonometric functions. Schwartz-Christoffel
transformation: mapping the interior of a polygon into the upper half-plane.
|
| 18. | Thursday, November 7 |
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. | Tuesday, November 12 |
HW 5 given.
Asymptotic expansion. Integration by parts. Laplace's method.
|
| 20. | Thursday, November 14 |
Watson lemma. Fourier-type integrals. Intermediate asymptotics.
|
| 21. | Tuesday, November 19 |
HW 6 given.
Method of steepest descents (saddle-point
method). Method of stationary phase.
Asymptotic evaluation of sums. Approximation by Riemann integral.
|
| 22. | Thursday, November 21 |
Euler-Maclaurin summation formula. Example: Stirling's formula.
Special functions. Introduction:
special functions and symmetries. Gamma function. Integral representations
by Euler and Hankel.
|
| 23. | Tuesday, November 26 |
Infinite product representations by Gauss and
Weierstrass. Properties of Gamma function. Asymptotic expansion of Gamma function.
function. Beta function.
Pochhammer's integral.
|
| | Thursday, November 28 |
No classes.
|
| 24. | Tuesday, December 3 |
HW 7 given.
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).
|
| 25. | Thursday, December 5 |
Modified Bessel functions
(I,K). 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. Generalized hypergeometric
functions.
|
| 26. | Tuesday, December 10 |
Basics of topology. Topology and
topological space. Homeomorphism. Topological properties.
Examples of topological spaces. Examples of topological spaces. Homotopy theory.
Zeroth homotopy group.
|
| 27. | Thursday, December 12 |
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. Examples
of higher homotopy groups for topological spaces.
|
| 28. | Tuesday, December 17 |
Optional lecture, 11:20am, in B-131.
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 19 |
Final exam.
11:00am - 1:30pm in P-128.
|