## Math 300: Introduction to Complex Variables

#### Location and Time

MWF 8:00-9:00 in BUCH A102

#### Textbook

We will use the textbook

Fundamentals of Complex Analysis with Applications to Engineering and Science (Third Edition), by E. Saff and A. Snider.

Homework problem sets will be from Saff & Snider. You can also refer to the free online textbook

A First Course in Complex Analysis by Beck, Marchesi, Pixton and Sabalka.

#### Topics

1.The complex numbers. (Saff & Snider sections 1.1 - 1.6)

• Definition of complex numbers, arithmetic operations.
• Geometry of complex numbers, modulus, conjugate.
• Basic inequalities: triangle and reverse triangle.
• Complex exponentials and the polar form of a complex numbers.
• The argument of a complex numbers.
• Trig identities via complex exponentials
• Geometry of multiplication
• Roots of unity, roots of a complex number.
• Planar sets

2.Analytic functions. (Saff & Snider sections 2.1 - 2.5)

• Complex functions
• Limits and conrinuity
• Complex differentiability and the Cauchy-Riemann equations.
• Analytic (Holomorphic) functions
• harmonic functions

3.Examples of Complex Functions. (Saff & Snider sections 3.1 - 3.3, 3.5)

• polynomials and rational functions
• complex exponential and trig functions
• complex logarithm, branches of the logarithm
• complex powers.

4.Complex Integration. (Saff & Snider sections 4.1 - 4.7)

• contours, definition and basic properties of complex integral.
• basic bound for complex integrals.
• computing integrals from the definition, fundamental theorem of calculus.
• equivalent conditions to existence of an antiderivative.
• Cauchy theorem
• Cauchy integral formula
• some consequences: Liouville's theorem and the fundamental theorem of algebra.
• maximum modulus principle for analytic functions.
• applications to harmonic functions: mean value property, maximum modulus principle for harmonic functions.
• Morera's theorem

5.Taylor and Laurent Series (Saff & Snider sections 5.1 - 5.6)

• sequences and series of complex numbers
• sequences and series of functions
• regions of convergence
• power series
• uniform convergence
• Taylor series, how to compute them.
• Laurent series
• zeros and isolated singularities, classification

6.The residue theorem (Saff & Snider sections 6.1 - 6.4)

• classification of singularities
• residues
• applications of the residue theorem
• argument principle and Rouche's theorem (time permitting)

There will be weekly homework assignments, usually due on Mondays. You will be required to upload files in pdf format (either scanned handwritten solutions, or prepared with a typesetting program such as LaTeX) on the course canvas page. A selection of problems will be graded. Late homework will not be accepted, even if you have a good excuse. To compensate, I will drop the lowest homework score. Even if you miss the deadline, its a good idea to do the problems, since this is the best way to prepare for the tests and exam. You are welcome to discuss the homework problems with your friends, but are expected to hand in your own work.

There will be two midterm exams on October 11 and November 8 in class and a final exam during the exam period.

 Homework (lowest score dropped): 10% Midterms: 2 x 20% Exam: 50%

If you miss a test for a legitimate reason, please fill out an academic request form and bring it to me. The weight of the test will be transfered onto the final exam.

#### Class notes, Homework assignments

To hand in homework, upload a pdf file on the canvas page before the due date.

 Date Reading Topics Problems Wed Sept 4 1.1 Introduction, arithmetic operations (Saff&Snider) 1.1: 6a, 8, 10, 20abc 30 due: Mon Sept 9 Fri Sept 6 1.2, 1.3 Modulus, conjugate . Mon Sept 9 1.4. 1.5 Triangle and reverse triangle inequatlities. Complex exponential, polar form of a complex number. (Saff&Snider) 1.2: 7def, 16, 17, 1.3:11,13,23, 1.4: 3c, 12b, 1.5 5ae 11 16 due: Mon Sept 16 Wed Sept 11 . Trig identity example: $(1+\cos(\theta) + \cos(2\theta)+\cdots +\cos(n\theta)$. The argument $\arg(z)$ and the principal argument $\mathop{Arg})$, geometry of multiplication . Fri Sept 13 1.6 Brief discussion of Julia sets. Here are some pictures: Julia sets, Roots of unity. Factorization of $1+z+z^2+\cdots + z^{n-1}$, roots of an arbitrary complex number $w$. . Mon Sept 16 2.1 Properties of sets in the plane, Complex functions intro (Saff&Snider) 1.6: 1ab, 2ab, 3ab, 4ab, 5ab, 6ab, 7ab, 8ab, 2.1:3abcd, 5abcd, 6abc, 10a, 11a due: Mon Sept 23

#### Files and Demos

Homework solutions: solutions1.pdf, solutions2.pdf

Geometry of complex multiplication

Here is a scan of the problems in Chapter 1. chapter1problems.pdf which you can use while the bookstore orders more copies of the text.

Here are some examples of describing sets with complex notation: setdescriptions.pdf.

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