12pm - 12:50pm on Mondays, Wednesdays, and Fridays in 140 Kaprielian Hall (KAP).
This course is an introductory to calculus in one complex variable. The main objects of study are functions of one complex variable which are holomorphic, i.e. complex differentiable. At first glance one might expect the whole theory to closely resemble ordinary calculus of a single real variable, but in fact holomorphic functions have many remarkable features not present in the real case.
We will assume familiarity with multivariate calculus.
There will be weekly problem sets. These will be listed on this website (see the schedule below) and will be handed in via Gradescope. Late homeworks will not be accepted without prior approval from the instructor.
Much of the course material will be developed in the problem sets. It is very important to do all of the problem sets to the best of your ability and to challenge yourself to solve the problems on your own, as this is the most effective way to absorb the material. We expect students to devote a significant amount of time to the problem sets.
While working on the problem sets, you are allowed to collaborate with your peers and to consult textbooks or other references. However, you must write down attributions for any peer, textbook, website etc from which you took any significant ideas. Moreover, you must attempt all problems on your own and your submitted solutions must be written out originally and individually. Submissions which are copied or suspiciously similar are subject to being rejected and potential disciplinary action.
| $\#$ | Date | Material | References | Problem set |
|---|---|---|---|---|
| 1 | Monday 1/13/25 | Introduction to the course. Complex numbers as pairs of real numbers and their multiplication formula. The complex plane, modulus of a complex number, complex conjugation. Finding square roots of complex numbers. | [BN] §1.1,§1.2,§1.3. | |
| 2 | Wednesday 1/15/25 | More on the complex plane. Polar form for complex numbers. Euler's formula. Finding squares using polar form. | [BC] §1 | Pset 1 (due Weds 1/22/25 at 11:59pm) |
| 3 | Friday 1/17/25 | Finding roots of unity and more general roots using polar form. Repeated roots of polynomials in terms of roots of the derivative. Roots of real polynomials come in complex conjugate pairs. Complex equations of half planes and circles. | [BC] §1. | |
| Monday 1/20/25: MLK day - no class | ||||
| 4 | Wednesday 1/22/25 | An complex equation for an ellipse. Basic topological notions: open sets, closed sets, limit points, accumulation points. Various examples. Limits of functions and continuity. | [BC] §1, §2. | Pset 2 (due Friday 1/31/25 at 11:59pm) |
| 5 | Friday 1/24/25 | More on topology: open and closed subsets, accumulation points, boundary points, closure of subsets, bounded subsets, compact subsets. Limits involving infinity. The triangle inequality. Reformulation of limits of complex valued functions in terms of real and imaginary parts converging. | [BC] §1, §2. | |
| 6 | Monday 1/27/25 | More on limits involving infinity and evaluating limits involving continuous functions. The extended complex plane and its identification with the Riemann two-sphere. The definition of holomorphic functions and various examples and non-examples, e.g. $f(z) = 17$, $f(z) = z$, $f(z) = z^2$, $f(z) = |z|$, $f(z) = \overline{z}$. | [BC] §1, §2. | |
| 7 | Wednesday 1/29/25 | Derivation of the Cauchy-Riemann equations. Formulation in terms of $u(x,y),v(x,y)$ and as $\partial_y f = i \partial_x f$. Verifying the Cauchy-Riemann equations in some simple examples. Formulation in terms of the Jacobian matrix of a function $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. Understanding complex numbers as $2 \times 2$ matrices and their corresponding linear transformations. | [BC] §2. | |
| 8 | Friday 1/31/25 | More on the Cauchy-Riemann equations. Pathological examples where CR holds but holomorphicity fails. Review of the mean value theorem. Proof that the Cauchy-Riemann imply holomorphic under extra technical assumptions. | [BC] §2 and [BN] §3. | Pset 3 (due Friday 2/7/25 at 11:59pm) |
| 9 | Monday 2/3/25 | More on properties of holomorphic derivatives (sum rule, product rule, quotient rule, chain rule). Polar form of the Cauchy-Riemann equations. Harmonic functions from holomorphic functions. Aside on Clairaut's theorem. Curves in the complex plane and line integrals. | [BC] §2 | |
| 10 | Wednesday 2/5/25 | Defining the elementary holomorphic: $e^z, \sin(z),\cos(z),\log(z)$. The principal argument and principal logarithm. | [BC] §3 | |
| 11 | Friday 2/7/25 | More on elementary functions. The principal value of $z^c$ and examples. Trigonometric and hyperbolic trigonometric functions. More on contour integrals and their independence under reparametrization. | [BC] §3, §4 | Pset 4 (due Friday 2/14/25 at 11:59pm) |
| 12 | Monday 2/10/25: asynchronous class | Some properties of contour integrals. Examples of contour integrals where the answer does or does not depend on the path. Upper bounds for contour integrals. The antiderivative theorem. | [BC] §4 | |
| 13 | Wednesday 2/12/25 | Review of contour integral properties and brief recap of examples. Proof of the antiderivative theorem. | [BC] §4 | |
| 14 | Friday 2/14/25 | Recap of the antiderivative theorem and some examples. Formulation of Cauchy's theorem. Proof in the case of continuous complex derivative using Green's theorem. | [BC] §4 | Pset 5 (due Friday 2/21/25 at 11:59pm) |
| Monday 2/17/25: President's Day: no class | ||||
| 15 | Wednesday 2/19/25 | Review of Cauchy's theorem and the proof in the continuously differentiable case using Green's theorem. Proof of Goursat's theorem (Cauchy's theorem for triangles). First example of a definite integral computation using Cauchy's theorem. | [BC] §4 | |
| 16 | Friday 2/21/25 | More detailed computation of the real definite integral $\int_0^\infty \frac{1-\cos(x)}{x^2}dx$ using Cauchy's theorem and contour integration. Introduction to Cauchy's integral formula. | [SS] Chapter 2 §3 and [BC] §4 | Pset 6 (due Friday 2/28/25 at 11:59pm) |
| 17 | Monday 2/24/25 | Proof of Cauchy's integral formula. Extension to derivatives, and in particular proof that analytic functions are infinitely complex differentiable. Statement of Liouville's theorem. Proof of the fundamental theorem of algebra via Liouville's theorem (to be finished next time). | [BC] §4 | |
| 18 | Wednesday 2/26/25 | Review of Cauchy's integral formula and its extension to derivatives. Cauchy's inequalities for derivatives. Proof of Liouville's theorem and the fundamental theorem of algebra. The maximum modulus principle and some examples and consequences. | [BC] §4 | |
| 19 | Friday 2/28/25 | Review of the maximum modulus principle. The mean value property for analytic functions and hence harmonic functions. Proof of the maximum modulus principle. | [BC] §4 | |
| 20 | Monday 3/3/25 | Filling gap in the proof of maximum modulus principle: an analytic function with constant norm must be (locally) constant. Introduction to complex power series. Review of the geometric series. The radius of convergence and Hadamard's formula. Power series for $e^z,\sin(z),\cos(z)$. | [BC] §5 | |
| 21 | Wednesday 3/5/25 | A little bit of midterm review. Proof that power series define analytic functions, and that analytic functions in a disk can be represented convergent power series. | [BC] §5 | |
| 22 | Friday 3/7/25 | Midterm in class | ||
| 23 | Monday 3/10/25 | Review of power series for analytic functions in a disc and computations of radii of convergence. Laurent series for analytic functions in an annulus. The order of vanishing for a zero of an analytic function. | [BC] §5 | Pset 7 (due Friday 3/14/25 at 11:59pm) |
| 24 | Wednesday 3/12/25 | Review of vanishing orders and some examples. Behavior of vanishing orders under various operatiobs. Observation that analytic functions have isolated zeros, and hence are determined by their values on arbitrarily small open subsets. Review of Laurent series and definition of residue. The residue formula and its proof. Examples. | [BC] §6 | |
| 25 | Friday 3/14/25 | Examples of residue computations and contour integrals using the residue formula. The residue of $\frac{1}{z-z_0}g(z)$ is just $g(z_0)$. Connection between the residue formula and Cauchy's integral formula. Computing real definite integrals (e.g. $\int_{-\infty}^{\infty} \tfrac{dx}{1+x^2}$). Introducing the trichotomy for isolated singularities: removal, pole, and essential. | [BC] §6 | |
| Monday 3/17/25: spring break - no class | ||||
| Wednesday 3/19/25: spring break - no class | ||||
| Friday 3/21/25: spring break - no class | ||||
| 26 | Monday 3/24/25 | More on using the residue formula to compute real definite integrals, e.g. $\int_{-\infty}^{\infty} \frac{1}{1+x^2}dx$ and $\int_{-\infty}^{\infty} \frac{e^{ax}}{1+e^x}dx$. More on the trichotomy for singularities of analytic functions. | [BC] §6 | |
| 27 | Wednesday 3/26/25 | Examples of removal singularities, poles, essential singularities, and non-isolated singularities. Formula for the residue of a pole in terms of derivatives. | [BC] §6 | Pset 8 (due Friday 4/2/25 at 11:59pm) |
| 28 | Friday 3/28/25 | The main theorem on removable singularities. Proof using a double keyhole contour argument. Characterization of poles in terms of $1/f$ having a removable singularity, and also in terms of $|f|$ limiting to infinity. | [BC] §6 | |
| 29 | Monday 3/31/25 | Review on removable singularities and poles. The Great Picard theorem. The Casorati-Weierstrass theorem and its proof. Introducing singularities and residue at infinity. | [BC] §6 | |
| 30 | Wednesday 4/2/25 | More on the residue at infinity. Geometric interpretation in terms of the Riemann sphere, and proof that it recovers the sum of residues inside of a contour. Example of a contour integral computation using the residue at infinity. | [BC] §6 | |
| 31 | Friday 4/4/25 | |||
| 32 | Monday 4/7/25 | |||
| 33 | Wednesday 4/9/25 | |||
| 34 | Friday 4/11/25 | |||
| 35 | Monday 4/14/25 | |||
| 36 | Wednesday 4/16/25 | |||
| 37 | Friday 4/18/25 | |||
| 38 | Monday 4/21/25 | |||
| 39 | Wednesday 4/23/25 | |||
| 40 | Friday 4/25/25 | |||
| 41 | Monday 4/28/25 | |||
| 42 | Wednesday 4/30/25 | |||
| 43 | Friday 5/2/25 | |||
| Final exam: Friday 5/9/25, 11am - 1pm |