# 
Date 
Read 
Topic 
1. 
Tuesday, September 2 

First meeting.
Introduction. Complex numbers.
Arithmetic operations.
Polar representation. de Moivre's
formula. Application to
sequences defined by recurrent relations. 
2.  Thursday, September 4 

HW 1 given.
Sequences and series. Power
series. Radius of convergence. etc. Elementary functions: exponent,
logarithm, power, trigonometric functions. Principal value of logarithm and
power. 
3.  Tuesday, September 9 

Geometrical properties of elementary functions: fractional
function. Stereographic projection. Square root. Riemann surface.
Geometrical properties of elementary functions: exponent, trigonometric functions.

4.  Thursday, September 11 

Geometrical properties of elementary functions: trigonometric functions.
Analytic functions.
Complex derivative. CauchyRiemann equations. CauchyRiemann equations
in polar form. 
5.  Thursday, September 11 

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 16 

HW 2 given.
Cauchy's theorem. Independence of integrals on contour in singlyconnected and
multiplyconnected domains of analyticity.
Cauchy's integral formula. Cauchy's formulas for
derivatives.

7.  Thursday, September 18 

Mean value theorem. Maximum modulus theorem.
Cauchy's inequalities and Liouville's theorem. Taylor series. 
8.  Thursday, September 18 

Laurent series.
Isolated singularities of singlevalued functions.
Removable
singularity, poles, and essential singularities. Singularities at infinity. 
9.  Tuesday, September 23 

Zeros and poles. Analytic continuation. Identity theorem. Singularity.
Monodromy theorem. Branching point singularity. Natural boundary.
Contour integration and residue calculus.
Residue theorem. 
10.  Thursday, September 25 

Residue at infinity. Examples of integrals. Symmetry trick. Logarithm trick.

11.  Thursday, September 25 

Logarithm trick. Jordan's Lemma.
Examples of integrals. 
 Tuesday, September 30 

NO CLASSES. Rosh Hashanah.

12.  Thursday, October 2 

HW 3 given. Examples of integrals. Contour shift trick.
Infinite series and infinite products. 
 Tuesday, October 7 

NO LECTURE (rescheduled).

 Thursday, October 9 

NO CLASSES. Yom Kippur.

 Tuesday, October 14 

NO LECTURE (rescheduled).

 Thursday, October 16 

NO LECTURE (rescheduled). 
 Tuesday, October 21 

NO LECTURE (rescheduled).

 Thursday, October 23 

NO LECTURE (rescheduled).

13.  Tuesday, October 28 

HW 4 given. (due Nov. 11) Infinite series and infinite products.
MittagLeffler's theorem. Examples of series and products.

14.  Thursday, October 30 

Conformal mapping.
Boundaryvalue problems. Dirichlet and von Neumann
boundaryvalue problems. Conjugate harmonic functions.
Twodimensional potential problems: hydrodynamics of an ideal incompressible liquid,
electrostatics and magnetostatics, heat flow.
Potential, stream function, and complex potential.

15.  Thursday, October 30 

Take Home Exam is given. (due Nov. 4)
Example: streamlining a dam.Invariance of Laplace equation under analytic
transformations. Generalized boundaryvalue problems (prescribed
singularities). Examples of boundaryvalue problems. Method of images.

16.  Tuesday, November 4 

Streamlining a cylinder. Conformal transformations.
Mapping of domains. Riemann's theorem.

17.  Thursday, November 6 

Doubly connected domains. Dirichlet problem on a circle. Poisson kernel. Examples of
conformal mappings. Bilinear transformation. Examples of
conformal mappings. Square root. Exponent.

18.  Tuesday, November 11 

Joukowsky transformation. Streamlining a plane wing. Blasius' theorem and Kutta condition.
Trigonometric functions. 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.

19.  Thursday, November 13 

HW 5 given.
Asymptotic methods. Example of asymptotic
expansion (exponential integral). Asymptotic expansions from the integration by parts.
The failure of the integration by parts.

20.  Thursday, November 13 

Laplace's method (examples). Watson lemma.

 Tuesday, November 18 

NO LECTURE (rescheduled).

21.  Thursday, November 20 

Fouriertype integrals. Laplace's method.
Intermediate asymptotics.
Method of steepest descents (saddlepoint
method).

22. 
Thursday, November 20 

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

23.  Tuesday, November 25 

HW 6 given.
Example: Stirling's formula. Special functions.
Introduction: special functions and symmetries.
Gamma function. Euler's formula.
Integral representations
by Euler and Hankel. Infinite product representations by Gauss and
Weierstrass. Properties of Gamma function.

 Thursday, November 27 

NO CLASSES. Thanksgiving.

24.  Tuesday, December 2 

Properties of Gamma function. Asymptotic expansion of Gamma function.
Digamma function. Beta function. Pochhammer's integral.

25.  Thursday, December 4 

Last year's final is given
(not for credit). The area of ddimensional 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). Modified Bessel functions
(I,K).

26.  Tuesday, December 9 

HW 7 given. 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.

27.  Thursday, December 11 

Generalized hypergeometric
functions. Orthogonal polynomials.
Example: Quantum oscillator and Hermite polynomials. Orthogonal polynomials corresponding
to an arbitrary weight function. Schmidt's process. Kernel. Formula of recurrence.

28.  Monday, December 15 

ChristoffelDarboux identity. Symmetry. Zeros. Least square property.
Differential equation. Rodriguez formula. Families of orthogonal polynomials and their applications.
Hermite, Laguerre, Legendre, Gegenbauer, Tchebyshev, Jacobi, etc. Vandermonde determinant and
integrals of Slater determinants.

Final. 
Tuesday, December 23 

Final exam.
8:00am10:30pm, P128 (Physics Building)
