Math 185 - Introduction to Complex Analysis

Instructor:

Christian Zickert

Office: Room 1053, Evans Hall.
Tel: +1 (510) 642-5546
email: zickert@math.berkeley.edu

Schedule:

Lectures: MWF 3:00-4:00pm, room 71 Evans.
Office Hours: Tu 10:00am-11:00am.

Textbook:

D. Sarason, Complex Function Theory, Second Edition, American Mathematical Society, 2007.

Grading:

Your grade will be determined as follows: 25% homework, 20% first midterm, 20% second midterm, 35% final exam.
The two lowest scoring sets of homework will not count.
Update: Since so many students had problems with the first midterm, this midterm will not count towards the grade.
The grade will thus instead be determined as follows: 35% homework, 25% second midterm, 40% final exam.

Course grades:

Your grades have been posted.
I hope you have learned a lot in this class, and I wish you all well in the future.

First midterm

The first midterm will take place Wednesday March 17 in class and will cover material up to and including VII.14.
The midterm will take place at Barrows 118.
No books or electronic aids are allowed. You may prepare one sheet of paper, listing essential results, to bring to the exam. Please contact me if you need special arrangements for the exam.
Using results from complex analysis, that we have not yet covered, will not receive credit.
Solutions to the first midterm.

Second midterm

The second midterm will take place Wednesday April 28 in class. It will cover material up to and including the lecture on Friday April 16. It will primarily focus on material covered since the first midterm, but knowledge of previous material will also be needed. Make sure to practise computation of integrals using the residue theorem. Note that the exercises VIII.12.1 and X.11.2 (covered in class) are part of the curriculum. As for the first midterm, no books or calculators are allowed, but you may prepare one sheet of paper, listing essential results, to bring to the exam. Here are a few practise problems: Compute the residue at 0 of sin(1/cos(1/z)); X.11.1, X.12.2, X.12.3, X.12.7, X.12.8. Please contact me if you need special arrangements for the exam.
Solutions to the second midterm.

Final exam

The final exam will take place Wednesday May 12, 7-10pm in room 71 Evans. Note that the exercises VIII.12.1 and X.11.2 (covered in class) are part of the curriculum. You are allowed to bring the textbook as well as your lecture notes and homework problems. Other textbooks are not allowed. The exam will contain at least one problem about computing a complicated integral using the residue theorem.
Solutions to the final exam.

Review session

There will be a review session on Wednesday May 5 at the usual time and place.

Homework:

Homework will be assigned and collected each Wednesday. Late homework will not be accepted.
Some notes by your grader.

Topics covered:

  • 01/20: A general introduction, basic properties of complex numbers (I 1-10).
  • 01/22: More basic properties, continuity.
  • 01/25: Holomorphic functions, the Cauchy Riemann equations.
  • 01/27: More on the Cauchy Riemann equations (II 5-8)
  • 01/29: Differential operators, conformality (II 9-10, 12-13).
  • 02/01: More on conformality (II 12-13).
  • 02/03: Harmonic functions (II 14-16), the extended complex plane (I 12-14).
  • 02/05: The Riemann sphere and complex projective space, linear fractional transformations (III 1-2).
  • 02/08: More on linear fractional transformations (III 3-5).
  • 02/10: Preservation of circles, examples (III 6-9, IV 1-3).
  • 02/12: Elementary functions (IV I-9).
  • 02/17: Branches of arg and log (IV 10-11), sequences of complex numbers.
  • 02/19: Sequences and series, tests for convergence (V 1-7).
  • 02/22: Power series, radius of convergence (V 8-14).
  • 02/24: Differentiation of power series (V 15-16).
  • 02/26: Cauchy product, path integration (V 17-18, VI 1-6).
  • 03/01: More on integration (VI 6-12).
  • 03/03: Versions of Cauchy's theorem (VII 1-3).
  • 03/05: Holomorphic functions have local power series expansions (VII 5-8).
  • 03/08: Liouville's theorem and the fundamental theorem of algebra, zeros of a holomorphic function (VII 11-13).
  • 03/10: The identity theorem, The Weierstrass convergence theorem (VII 14-15).
  • 03/12: The maximum modulus principle, Schwarz's Lemma, harmonic functions (VII 16-18).
  • 03/15: Review for the midterm.
  • 03/17: Midterm.
  • 03/19: Laurent series (VIII 1-2).
  • 03/22-03/26: Spring break.
  • 03/29: More on Laurent series, isolated singularities (VIII 3-7).
  • 03/31: Behaviour of a function near an isolated singularity (VIII 8-9).
  • 04/02: Logarithms and arguments along a curve (IX 1-3).
  • 04/05: Winding numbers and contours (IX 4-7).
  • 04/07: Cauchy's theorem (IX 8-10).
  • 04/09: Homotopy, simply connected domains (IX 11-14, X 1-5).
  • 04/12: Residues and the residue theorem (VIII 12, X 7-8, 10).
  • 04/14: Computation of integrals using the residue theorem (X 10).
  • 04/16: The argument principle, Rouché's theorem (X 11, 12).
  • 04/19: The local mapping theorem and consequences (Section X 13-15).
  • 04/21: Conformal equivalence, examples (X 16).
  • 04/23: Maps into the unit disk, the Stieltjes-Osgood Theorem (X 17-19).
  • 04/26: Review for the midterm.
  • 04/28: Midterm.
  • 04/30: The Riemann mapping theorem (X 20-21).