## ANNOUNCEMENTS

Registration for Spring term opens Thursday, February 23, 2016.

This page was last updated: .

## COURSE DESCRIPTION

The course will be an introduction to complex analysis. The following topics will be covered: Holomorphic functions and the Cauchy-Riemann equations, Cauchy's theorem and Cauchy's integral formula, Taylor expansions, entire functions and Liouville's theorem, zeros of holomorphic functions, isolated singularities and Laurent expansions, meromorphic functions, the Residue Theorem, the Maximum Modulus Principle, conformal mappings and linear fractional transformations, harmonic functions, infinite products and the Weierstrass Factorization Theorem, the Gamma function, and the prime number theorem.

## INSTRUCTORS

Farzad Fathizadeh

OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS

358 Sloan

626-395-4355

farzadf@caltech.edu

## POLICIES

### Grades

Based on homework (70%), and final exam (30%).

### Homework Policy

Collaboration for solving the homework problems is allowed, however the students should write up the solutions individually. The homework sets should be handed in with a cover sheet that only has the student's first and last names on it, due to privacy policies.

## TEXTBOOKS

Elias M. Stein and Rami Shakarchi,
*
Complex Analysis (Princeton Lecture Series in Analysis II),
*
ISBN-13: 978-0-691-11385-2.

## LECTURE NOTES

Date | Description |
---|---|

Mon April 3rd | Holomorphic functions, Cauchy-Riemann equations |

Wed April 5th | Holomorphicity from the Cauchy-Riemann equations, power series |

Fri April 7th | Radius of convergence, complex differentiabilty of power series, smooth curves |

Monday April 10th | Integral of complex functions over curves, primitives, integral of functions with perimitives |

Wed April 12th | Examples of integrals of curves, logarithm of complex numbers and its branches, statmenet of Goursat's theorem |

Fri April 14th | Proof of Goursat's theorem, idea of existence of primitives for holomorphic functions on a disk |

Mon April 17th | Existence of primitives for holomorphic functions on the interior of toy contours, Cauchy's theorem |

Wed April 19th | Evaluation of some integrals using Cauchy's theorem, statment of the Cauchy integral formula |

Fri April 21st | Cauchy's integral formula, Cauchy inequalitites, Liouville's theorem, fundamental theorem of algebra |

Mon April 24th | Power series expansion for holomorphic functions, identically vanishing of holomorphic functions whose set of zeros has a limit point, Morera's theorem, closedness of the set of holomorphic functions under uniform convergence on compact sets |

Wed April 26th | zeros and sigularities of holomorphic functions, poles, order of zeros and poles, residues and the residue formula |

Fri April 28th | Examples of integration using the residue formula |

Mon May 1st | Riemann's theorem on removable singularities, Casorati-Weierstrass theorem, meromorphic functions |

Wed May 3rd | Meromorphic functions on the extended plane and the proof that they are the rational functions, Riemann sphere and the stereographic projection, the argument principle |

Fri May 5th | Rouche's theorem: stability of the number of zeros of a holomorphic function inside a closed curve under small enough perturbations, the open mapping theorem, the maximum modulus principle, homotopies |

Mon May 8th | Integrals of homolomorphic function over homotopic curves, simply connected regions |

Wed May 10th | Existence of primitives for holomorphic functions on simply connected regions, the branch of the logarithm associated with a simply connected region, the pricipal branch of the complex logarith and its power series expansion |

Fri May 12th | The logarithm of a nowhere vanishing holomorphic function on a simply conneted region, Fourier series and the Taylor coefficients of holomorphic functions, the mean-value property |

Mon May 15th | Infinite products and holomorphic functions, the product formula for the sine function, properties of the cotangent function |

Wed May 17th | Proof of the product formula for the sine function, Weierstrass infinite products and canonical factors |

Fri May 19th | Weierstrass' theorem on exsitence of entire functions with prescribed zeros with desired multiplicities, Hadamard's factorization theorem |

Mon May 22nd | The gamma function and properties of its meromorphic extension to the plane, zeros of the multiplicactive inverse of the Gamma function as an entire function |

Wed May 24th | Order of growth of the multiplicative inverse of the Gamma function and its Hadamard factorization, the Riemann zeta function and properties of its meromorphic extension to the complex plane using the Mellin transform and the generating function of the Bernoulli numbers |

Fri May 26th | Ditch day |

Mon May 29th | Holiday |

Wed May 31st | The functional equation for the Riemann zeta function, the theta function, Poisson summation formula |

Fri June 2nd | The product formula for the Riemann zeta function, zeros of the zeta function and absence zeros on the line \( \Re(s) =1 \), the statement of the prime number theorem |

Mon June 5th | Conformal maps and conform equivalence, a conformal map from the disc into the upper half-plane, further examples of conformal maps |

Wed June 7th | The Schwarz lemma, automorphisms of the disc and upper half-plane, statement of the Riemann mapping theorem |

Fri June 9th | Proof of the Riemann mapping theorem |

## HOMEWORK

Due Date | Homework | Solutions |
---|---|---|

4 p.m., Thursday April 13th | Chapter 1 of the textbook, exercises 7, 8, 9, 12, 16, 19 | |

4 p.m., Thursday April 20th | Chpater 1: exercises 10, 11, 25; Chapter 2: exercise 5 and problem 1 | |

4 p.m., Thursday April 27th | Chapter 2: exercises 1, 2, 6, 7, 11, 12 | |

4 p.m., Thursday May 4th | Chapter 2: exercise 3; Chpater 3: exercises 2, 5, 8 and problem 3 | |

4 p.m., Thursday May 11th | Chapter 3: Exercises 10, 12, 14, 17 | |

4 p.m., Thursday May 18th | Chapter 3: Exercises 18, 20, 22 and problem 2 | |

4 p.m., Thursday May 25th | Chapter 5: Exercises 6, 8, 9, 10, 11 | |

4 p.m., Friday June 2nd | Chapter 6: Exercises 4, 5, 7, 8, 10 |

## EXAMS

Your final exam is available here. Please refer to your email for the password. If you have any issues, please contact Meagan Heirwegh.