# 
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. CauchyRiemann equations. CauchyRiemann equations
in polar form. 
5.  Tuesday, September 19 

CauchyRiemann equations
in complex form. Harmonic functions. Integration in
complex plane. Independence of integrals on contour in singlyconnected and
multiplyconnected 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 singlevalued 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. MittagLeffler's theorem. Examples of series and products.

13.  Tuesday, October 10 

HW 4 given.
Conformal mapping.
Boundaryvalue problems. Dirichlet and von Neumann
boundaryvalue problems. Conjugate harmonic functions.
Twodimensional potential problems: electrostatics and magnetostatics, heat flow.
Potential, stream function, and complex potential.

14.  Tuesday, November 7 

Twodimensional potential problems: hydrodynamics of an ideal
incompressible liquid. Example:
streamlining a dam. Invariance of Laplace equation under analytic
transformations. Generalized boundaryvalue problems (prescribed
singularities).

15.  Tuesday, November 7 

Examples of boundaryvalue 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. SchwartzChristoffel
transformation: mapping the interior of a polygon into the upper halfplane.

18.  Tuesday, November 14 

SchwartzChristoffel
transformation: mapping the exterior of a polygon into the exterior
of a unit circle. Causality and KramersKronig (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. Fouriertype integrals. Laplace's method. Intermediate asymptotics.
Method of steepest descents (saddlepoint
method).

21.  Tuesday, November 21 

Method of steepest descents (saddlepoint
method). Method of stationary phase. Asymptotic evaluation of sums. Approximation by Riemann integral.
EulerMaclaurin 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 ddimensional 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 B131.
