Ma 108c:  Classical Analysis (Spring 2016-17)

ANNOUNCEMENTS

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

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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.


PREREQUISITES

Ma 1 or equivalent, or instructor's permission.


SCHEDULE

Monday, Wedensday, Friday, 11:00 - 11:55 a.m., 151 Sloan.


INSTRUCTORS

Farzad Fathizadeh
OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS
358 Sloan
626-395-4355
farzadf@caltech.edu


TA's

Connor Meehan
156 Sloan
cgmeehan@caltech.edu


OFFICE HOURS

Instructor: Fridays, 3 to 4 p.m., Sloan 358.
TA: Wednesdays, 5 to 6 p.m., Sloan 159


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.


TOPICS COVERED

Please refer to the Course Description.


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
Fri June 2nd
Mon June 5th
Wed June 7th
Fri June 9th

HOMEWORK

Due Date Homework Solutions
4 p.m., Thursday April 13th Chapter 1 of the textbook, exercises 7, 8, 9, 12, 16, 19 Hints 1
4 p.m., Thursday April 20th Chpater 1: exercises 10, 11, 25; Chapter 2: exercise 5 and problem 1 Hints 2
4 p.m., Thursday April 27th Chapter 2: exercises 1, 2, 6, 7, 11, 12 Hints 3
4 p.m., Thursday May 4th Chapter 2: exercise 3; Chpater 3: exercises 2, 5, 8 and problem 3 Hints 4
4 p.m., Thursday May 11th Chapter 3: Exercises 10, 12, 14, 17 Hints 5
4 p.m., Thursday May 18th Chapter 3: Exercises 18, 20, 22 and problem 2 Hints 6
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

 


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