Complex Variables

Week 1

(no class)

• Syllabus; basics of complex numbers
• The corresponding homework.

 
• Review of Taylor theorem and Taylor series (Calculus II)
• The corresponding homework.

Week 2

 
• Review of power series and convergence tests; the concept of a real analytic function
• The corresponding homework.

• Practice
• The corresponding homework.

• Practice
• The corresponding homework.

Week 3

 
• Conjugates, moduli and misc. other stuff about the complex plane.
• The corresponding homework.

• Sequences and series of complex numbers
• Homework POSTPONED.

• Analytic function of a complex variable
• The corresponding homework.

Week 4

• The exponential function
• The corresponding homework.

 
• Trigonometric functions; foundations of numbers e and Pi
• The corresponding homework.

• More on the number Pi
• The corresponding homework.

Week 5

 
• Towards the concept of a logarithmic function
• The corresponding homework.

 
• The concept of a branch cut; logarithmic function
• The corresponding homework.

 
• Power function and roots of complex numbers
• The corresponding homework.
• The STUDY GUIDE for the upcoming EXAM!

Week 6

 
• The definition of the derivative; holomorphic functions

Practice day!

EXAM!!! (The Study Guide)

Week 7

 
• Analytic functions are holomorphic
• The corresponding homework

 
• Cauchy-Riemann equations
• The corresponding homework

 
• Contour integration
• The corresponding homework

Week 8

 
• More on contour integration; estimating contour integrals
• The corresponding homework

• From the Fundamental Theorem of Calculus to Cauchy Theorem
• The corresponding homework

• Cauchy Theorem(s)
• The corresponding homework

Week 9

• Talyor series; holomorphic functions are analytic
• The corresponding homework

• Synthesis day
• The corresponding homework

• Liouville Theorem, The Fundamental Theorem of Algebra
• The corresponding homework

Week 10

• Identity Theorem; the concept of analytic continuation
• The corresponding homework

• Laurent series
• The corresponding homework

• Practice
• The corresponding homework

Week 11

• Residues and poles
• The corresponding homework

• Integration using residue theory
• The corresponding homework

• Practice
• The corresponding homework

Week 12

• Practice
• No new homework, but here is the EXAM STUDY GUIDE

• Practice
• EXAM is NEXT Thursday (7pm, Howard 254)

• Integral transforms

Week 13

• More on integral transforms

• The Gamma function

• Zeta function

Week 14

• Zeta function
• The corresponding homework.
• See also the Final Exam Study List.

• On the distribution of prime numbers

(none)