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
Wed April 26th
Fri April 28th
Mon May 1st
Wed May 3rd
Fri May 5th
Mon May 8th
Wed May 10th
Fri May 12th
Mon May 15th
Wed May 17th
Fri May 19th
Mon May 22nd
Wed May 24th
Fri May 26th
Mon May 29th
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  
4 p.m., Thursday April 20th Chpater 1: exercises 10, 11, 25; Chapter 2: exercise 5 and problem 1 (textbook)  
4 p.m., Thursday April 27th Chapter 2: exercises 1, 2, 6, 7, 11, 12  
     
     
     
     
     

EXAMS

 


READING