MATH50001 Analysis 2 (Complex Analysis) 2021
Ari Laptev
Course information
The main goal of this course is to give an introduction to basic facts of Theory Complex functions.
Tutorials, January 21 - March 19:
Mondays and Tuesdays 11:30-12:20.
Q&A January 25 - March 19:
Thursdays 11:00-11:40 and Fridays 16:00-16:40.
Course work 1 - Friday, February 19.
Mid-term test: Wednesday, February 24.
Course work 2 - Friday, March 26.
Recommended Schedule:
January 11-15 - Lectures 1
January 18-22 - Lectures 2-3
January 25-29 - Lectures 4-5 + Problem sheet 1
February 1-5 - Lecture 6-7 + Problems sheet 2
February 8-12 - Lectures 8-9 + Problem sheet 3
February 15-19 - Lectures 10-11 + Problem sheet 4
CW1 (Friday, February 19) will include only the material from Lectures 1-9 only
February 22-26 - Lecture 12
Mid-term test (Wednesday, February 24) will include the material from Lectures 1-11 only.
March 1-5 - Lectures 13-14 + Problem sheet 5
March 8-12 - Lectures 15-16 + Problem sheet 6
March 15-19 - Lectures 17-18 + Problem sheet 7
March 22-26 Lectures 19-20
CW2(Friday, March 26) will include the material from Lectures 12 - 18.
Content
Holomorphic Functions: Definition using derivative, Cauchy-Riemann equations, Polynomials, Power series,
Rational functions, Moebius transformations.
Cauchy's Integral Formula: Complex integration along curves, Goursat's theorem, Local existence of primitives and Cauchy's theorem in a disc, Evaluation of some integrals, Homotopies and simply connected domains, Cauchy's integral formulas.
Applications of Cauchy's integral formula: Morera's theorem, Sequences of holomorphic functions, Holomorphic functions defined in terms of integrals, Schwarz reflection principle. Meromorphic Functions: Zeros and poles. Laurent series. The residue formula, Singularities and meromorphic functions, The argument principle and applications, The complex logarithm.
Harmonic functions: Definition, and basic properties, Maximum modulus principle. Conformal Mappings: Definitions, Preservation of Angles, Statement of the Riemann mapping theorem.
Lectures
Recorded lectures could be found here. Please click on "Complex amalysis" (so far Lectures 1-16)
Lecture 1 (pdf)
Lecture 2 (pdf)
Lecture 3 (pdf)
Lecture 4 (pdf)
Lecture 5 (pdf)
Lecture 6 (pdf)
Lecture 7 (pdf)
Lecture 8 (pdf)
Lecture 9 (pdf)
Lecture 10 (pdf)
Lecture 11 (pdf)
Lecture 12 (pdf)
Lecture 13 (pdf)
Lecture 14 (pdf)
Lecture 15 (pdf)
Lecture 16 (pdf)
Problems:
probl.1
( sol.1 )
probl.2
( sol.2 )
probl.3
( sol.3 )
probl.4
( sol.4 )
probl.5
( sol.5 )
probl.6
( sol.6 )
probl.7
( sol.7 )
If necessary please contact me either via Teams, e-mail or Skype. My skype name is: arilaptev
Recommended Student Texts:
Barry Simon,
A Comprehensive Course in Analysis,
Part 2A: Basic Complex Analysis,
American Math Society, 2015.
Elias M. Stein & Rami Shakarchi,
II Complex Analysis,
Princeton University Press, 2003.
Elias M. Stein & Rami Shakarchi,
I Fourier Analysis,
Princeton University Press, 2003.
John M. Howie,
Complex Analysis,
Springer, 2007.
Walter Rudin,
Real and Complex Analysis,
2nd ed., McGraw-Hill, 1974.
Lars V. Ahlfors,
Complex Analysis,
3rd ed., McGraw-Hill, 1979.