# 
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. CauchyRiemann equations. CauchyRiemann equations
in polar form. 
5.  Thursday, September 20 

HW 2 given. 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 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 singlevalued 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 

MittagLeffler's theorem. Examples of series and products. Conformal mapping.
Boundaryvalue problems. Dirichlet and von Neumann
boundaryvalue problems. 
13.  Thursday, October 18 

Take Home Exam is given. Conjugate harmonic functions.
Twodimensional potential problems: electrostatics and magnetostatics, heat flow.
Potential, stream function, and complex potential. Twodimensional 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 boundaryvalue problems (prescribed
singularities). Examples of boundaryvalue 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 

SchwartzChristoffel
transformation: mapping the interior of a polygon into the upper halfplane,
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 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. Fouriertype integrals. Laplace's method. Intermediate asymptotics.
Method of steepest descents (saddlepoint
method).

21.  Thursday, November 15 

Method of steepest descents (saddlepoint
method). Method of stationary phase.

22. 
Tuesday, November 20 

HW 6 given. Asymptotic evaluation of sums. Approximation by Riemann integral.
EulerMaclaurin 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 ddimensional 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. 10am12pm, B131. 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.
9:30am12:00pm, B131 (Physics Building)
