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Schedule
Notes
The schedule is tentative and will change as the course progresses.
For information about add/drop dates and other matters related to course registration, please consult the Registrar's Website.
All dates and times are in ET (Eastern Time).
Section numbers are as in Complex Variables (2nd Edition) by Stephen D. Fisher.
There will usually be one homework assignment due on Saturday every week.
Week of January 12th (Week 1)
Tuesday, January 13th
Topics: the complex numbers, \(\mathbb{C}\), and their construction; the algebra of \(\mathbb{C}\); modulus, conjugate, real part, and imaginary part.
Reading: §1.1 The Complex Numbers and the Complex Plane.
Video: (Kaltura01)
Thursday, January 15th
Topics: the fundamental theorem of algebra; the geometry of \(\mathbb{C}\) via the Argand plane; the geometry of addition, conjugation, and magnitude; polar coordinates and the geometry of multiplication; de Moivre's theorem and its corollaries.
Reading: §1.1 The Complex Numbers and the Complex Plane; §1.2 Some Geometry.
Video: (Kaltura02)
Note: due to a technical issue with the webcam, the Kaltura02 recording only begins halfway through class.
Week of January 19th (Week 2)
Tuesday, January 20th
Topics: the triangle inequality; equations for lines and circles in \(\mathbb{C}\); open disc; interior point, exterior point, and boundary point; the boundary of a set; open sets.
Reading: §1.2 Some Geometry; §1.3 Subsets of the Plane.
Video: (Kaltura03)
Thursday, January 22nd
Topics: quick review of last class; proving a set is open; examples of open sets; closed sets; \(\mathbb{C}\) and the empty set are clopen; \(U\) is open if and only if \(\mathbb{C} \setminus U\); polygonal paths; connected sets.
Reading: §1.3 Subsets of the Plane.
Video: (Kaltura04)
Week of January 26th (Week 3)
HW 2 is due on Tuesday, February 3rd at 11 PM (note: unusual day!).
I will be out of town for part of this week; please see below for necessary adjustments.
No Office Hours on Tuesday this week (in view of my travel).
Tuesday, January 27th
Online Class (via Zoom).
Topics: quick review of last class; convex sets; convex open sets are connected; functions; domain, co-domain, and range; computing range; surjective, injective, and connections to invertibility; an explicit fractional linear transformation and its action on the unit disc; visualizing complex functions; \(\mathbb{C}\) and the empty set are clopen; \(U\) is open if and only if \(\mathbb{C} \setminus U\); polygonal paths; connected sets.
Reading: §1.3 Subsets of the Plane; §1.4 Functions and Limits.
Video: (Kaltura05)
Thursday, January 29th
Topics: sequences; limit of a sequence; reducing limits of complex sequences to limits of real sequences; closure of a set; limit of a function; stereographic projection; the Riemann sphere \(\hat{\mathbb{C}} = \mathbb{C} \cup \{ \infty\}\).
Reading: §1.4 Functions and Limits.
Video: (Kaltura06)
Week of February 2nd (Week 4)
HW 2 is due on Tuesday, February 3rd at 11 PM (note: unusual day!).
HW 3 is due on Saturday, February 6th at 11 PM.
Tuesday, February 3rd
Topics: quick review of last class; limit of a function at infinity; continuity of functions; continuity of polynomials and rational functions; continuity at infinity; infinite series; partial sums; reducing complex series to real series; absolute convergence.
Reading: §1.4 Functions and Limits.
Video: (Kaltura07)
Thursday, February 5th
Topics: quick review of last class including previously omitted proofs; the geometric series and convergence thereof; quick review of standard calculus tests; the exponential function; Euler's formula as a definition; \(e^{z+w} = e^z \cdot e^w\).
Reading: §1.4 Functions and Limits; §1.5 The Exponential, Logarithm, and Trignometric Functions.
Video: (Kaltura08)
Week of February 9th (Week 5)
HW 4 is due on Saturday, February 14th at 11 PM.
I will be out of town for part of this week; please see below for necessary adjustments.
No Office Hours on Tuesday this week (in view of my travel).
Tuesday, February 10th
Topics: mapping properties of the exponential functions; the logarithm; the principal branch of the logarithm; constructing other branches of the logarithm; exponentiation of the form \(a^z\) when \(a, z \in \C\) as a multi-valued function.
Reading: §1.5 The Exponential, Logarithm, and Trignometric Functions.
Video: (Kaltura09)
Thursday, February 12th
Review for Midterm 1.
Topics: trigonometric functions; inverse trigonometric functions.
Reading: §1.5 The Exponential, Logarithm, and Trignometric Functions.
Video: (Kaltura10)
Week of February 16th (Week 6)
Midterm 1 is on Monday, February 16th from 8:00 PM to 9:00 PM in LWSN B155.
HW 5 is due on Saturday, February 21st at 11 PM.
Extra Office Hours on M, 4:00 PM to 6:00 PM this week (in addition to usual Office Hours) in view of Midterm 1.
Tuesday, February 17th
Topics: complex differentiability and derivatives; analytic or holomorphic functions; entire functions; basic examples of holomorphic functions; rules of differentiation; differentiability implies continuity; the Cauchy-Riemann equations; harmonic functions.
Reading: §2.1 Analytic and Harmonic Functions; the Cauchy–Riemann Equations.
Video: (Kaltura11)
Thursday, February 19th
Topics: the connection between holomorphic functions and harmonic functions; harmonic conjugates; finding harmonic conjugates; the equivalence of holomorphicity of \(f = u + iv\) with the harmonic conjugacy of \(u\) and \(v\); the branch of a logarithm as a holomorphic function; some corollaries of the Cauchy-Riemann equation.
Reading: §2.1 Analytic and Harmonic Functions; the Cauchy–Riemann Equations.
Video: (Kaltura12)
Week of February 23rd (Week 7)
Tuesday, February 24th
Thursday, February 26th
Week of March 2nd (Week 8)
Tuesday, March 3rd
Thursday, March 5th
Week of March 9th (Week 9)
Tuesday, March 10th
Thursday, March 12th
Week of March 16th (Week 10)
Tuesday, March 17th
Thursday, March 19th
Week of March 23rd (Week 11)
HW 9 is due on Saturday, March 27th at 11 PM.
I will be out of town for part of this week; please see below for necessary adjustments.
No Office Hours on Tuesday this week (in view of my travel).
Tuesday, March 24th
Thursday, March 26th
Week of March 30th (Week 12)
Midterm 2 is on Monday, March 30th from 8:00 PM to 9:00 PM in LWSN B155.
Extra Office Hours on M, 4:00 PM to 6:00 PM this week (in addition to usual Office Hours) in view of Midterm 2.
Tuesday, March 31st
Thursday, April 2nd
Week of April 6th (Week 13)
Tuesday, April 7th
Thursday, April 9th
Week of April 13th (Week 14)
HW 12 is due on Saturday, April 18th at 11 PM.
I will be out of town for part of this week; please see below for necessary adjustments.
No Office Hours on Tuesday this week (in view of my travel).
Tuesday, April 14th
Thursday, April 16th
Week of April 20th (Week 15)
Tuesday, April 21st
Thursday, April 23rd
Week of April 27th (Week 16)
Tuesday, April 28th
Thursday, April 30th
Week of May 4th (Final Exam Week)
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