This course
will focus on a topic typically covered in a second course in complex
analysis: Elliptic function theory. Elliptic functions as developed by Jacobi, Weierstrass, Eisenstein, Dedekind, and others are one of the crowning achievements
of 19th century mathematics and are widely applied in physics and engineering.
Their study is the natural continuation of the analysis of polynomial,
exponential, and trigonometric functions of a complex variable. In the 20th
century, the analysis of the beautiful transformation properties of elliptic
functions developed into the theory of elliptic curves and modular forms, a
central topic of algebraic geometry and number theory. Recently, elliptic
functions have played an important role in the 21st century mathematics
inspired by theoretical physics.

Here are some topics from the course:

Arithmetic-Geometric
Mean (AGM) and elliptic integrals;

Doubly
periodic q-functions
Jacobi;

The
Weierstrass P-function;

Applications to PDEs (maximal
surfaces equation) and number theory (representing integers as sums of squares
and quadratic forms).

### Literature

__Akheizer____, N.I.__ *Elements of the Theory of Elliptic
Functions.* Providence, RI: Amer. Math. Soc.,
1990.

__Chandrasekharan____, K.__ *
**
Elliptic Functions.
* Berlin: Springer-Verlag, 1985.

__Hille____, E.__ *
Analytic Function Theory*, Vol. II. Ginn
& Co., 1962.

### Lecture notes

here

### Evaluation

*TBA*

### Other Information

Here you can find
the list of basic and advanced books in Elliptic Functions and the related
topics

Here is the Wolfram Research documentation concerning Elliptic functions
and Elliptic Integrals for the *Mathematica* users

The
list of the Jacobi Elliptic Functions formulae is here

Applications:
Audio Synthesis by Means
of Elliptic Functions *due to* Vittorio Cafagna and Filippo D'Eliso

This may also be useful J

and
more