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SYLLABUS AND HOMEWORK

Mathematics 4450 and 5450

Introduction to Complex Variables

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Homework is due in class and must be stapled, with your name on it, to receive credit.

You may find the Mathematics Help Room (MATH 175) to be useful as a meeting point for discussing homework.

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Date	Topics covered	Reading	Homework Assignments
Monday January 12	Review of complex numbers	Chapter 1
Wednesday January 14	Complex exponential, powers and roots.
Friday January 16	The Riemann sphere and stereographic projection.		HW1: Due Friday Feburary 6 Chapter 1 Exercises: Section 1.1 #2,15,26 Section 1.2 #1,4,8,20,21 Section 1.3 #2,5,22 Section 1.4 #3,9,23 Section 1.5 #4,17 Section 1.6 #2-8 Recommended (not for credit): Section 1.6 #9-14, 16-18
Monday January 19	NO CLASS		NO CLASS
Wednesday January 21	Holomorphic functions Functions of a complex variable, limits and continuity, holomorphicity.	Chapter 2
Friday January 23	The Cauchy--Riemann equations.		HW 2: Due Friday February 6 Chapter 2 Exercises: Section 2.1 #1(a),(b),5,12,14,15 Section 2.2 #1,2,3,5,9,18,20,22 Section 2.3 #2,4,6,7(b),14,16 Section 2.4 #1,2,4,10,15
Monday January 26	Elementary functions Polynomials and rational functions, the exponential function.	Chapter 3
Wednesday January 28	Trigonometric and hyperbolic functions, the logarithmic function, washers, wedges and walls.
Friday January 30	Complex powers and inverse trigonometric functions, applications.		HW 3: Due Friday February 6 Chapter 3 Exercises: Section 3.1 #2,3(a),4,7,11,14 Section 3.2 #3,7,14,18 Section 3.3 #8,9,11 Section 3.4 #1,2,3 Section 3.5 #1,5,6,11
Monday February 2	Analytic functions Sequences and series, Taylor series, Power series.	Chapter 5
Wednesday February 4	Convergence, Laurent series, zeros and singularties.	Read the following pdf on convergence.
Friday February 6	The point at infinity, analytic continuation.		HW 5: Due Friday February 20 Chapter 5 Exercises: Section 5.1 #1,5,6,13,16 Section 5.2 #1,2,5,10,13 Section 5.3 #1,2,3,10,12,13 Section 5.5 #1,3,9 Section 5.6 #1,2,5,6,9
Monday February 9	Homework review
Wednesday February 11	Review
Friday February 13	MIDTERM I		MIDTERM I
Monday February 16	Complex integration. Contours, contour integrals, independence of path.	Chapter 4	HW 4: Due Friday February 27 Chapter 4 Exercises: Section 4.1 #2,3,11 Section 4.2 #2,3,8,12,15 Section 4.3 #1,2,3,4,5,10,11,12 Do the exercises on this pdf. Recommended (not for credit): Section 4.4 #1,3,9,11,15 Section 4.5 #1,2,3,6,9,10,11 Section 4.6 #3,4,5,6,12,14,15 Section 4.7 #1,2,3,4,5,6,7,8,9
Wednesday February 18	Cauchy's integral theorem, Cauchy's integral formula and its consequences.
Friday February 20	Bounds for holomorphic functions, applications.
Monday February 23	Reside theory Residue theorem, trigonometric integrals.	Chapter 6
Wednesday February 25	Improper integrals.
Friday February 27	Improper integrals involving trigonometric functions.		HW 6: Due Friday March 6 Chapter 6 Exercises: Section 6.1 #1,2,3,7 Section 6.2 #1,2,3 Section 6.3 #1,2,3,8
Monday March 2	Indented contours.
Wednesday March 4	Integrals involving multiple-valued functions.
Friday March 6	The argument theorem and Rouche's theorem.		HW 7: Due Friday April 3 Chapter 6 Exercises: Section 6.4 #1,2,3 Section 6.5 #3,4,5 Section 6.7 #1,2,4,5,6 Recommended (not for credit): Section 6.3 #14,15,18 Section 6.7 #13,14,18
Monday March 9	Conformal mapping	Chapter 7
Wednesday March 11	Harmonic functions. Invariance of Laplace's equation.
Friday March 13	Geometric considerations.		HW 8: Due Friday April 10 Chapter 2 Exercises: Section 2.5 #1,5,9,10,20,21 Chapter 7 Exercises: Section 7.1 #1,2,3 Section 7.2 #1,2,3,4,5,10,11,14
Monday March 16	Homework review
Wednesday March 18	Review
Friday March 20	MIDTERM II
Week of March 23-March 27	SPRING BREAK		SPRING BREAK
Monday March 30	Mobius transformations.
Wednesday April 1	The Schwartz--Christoffel transformation.
Friday April 3	Applications.		HW 8: Due Friday April 17 Chapter 7 Exercises: Section 7.3 #1,2,3 Section 7.4 #5,10,11,12,21,22 Section 7.5 #1,3,4
Monday April 6	Transforms in complex analysis Fourier series.	Chapter 8
Wednesday April 8	Fourier transforms.
Friday April 10	Fourier transforms continued, and Laplace transforms.		HW 9: Due Friday April 24 Chapter 8 Exercises (not to be turned in)
Monday April 13	Laplace transforms continued.
Wednesday April 15	z-transforms.
Friday April 17	Cauchy integrals and the Hilbert transform.		HW 10: Due Friday April 24 Chapter 8 Exercises (not to be turned in)
Monday April 20	Further topics
Wednesday April 22	Further topics
Friday April 24	Further topics		HW 11: Due Friday May 1 Study for final exam
Monday April 27	Further topics
Wednesday April 29	Homework review
Friday May 1	Review
Thursday May 7	FINAL EXAM 1:30 - 4:00 PM ECCR 118 (Lecture Room)	FINAL EXAM	FINAL EXAM

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I strongly encourage everyone to use LaTex for typing their homework. If you have a mac, one possible easy way to get started is with texshop. If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc. If you are using windows, you're on your own, but I'm sure there's something online. Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file). It is probably easiest to just look at the .tex file, and start to experiment. There is also the (Not so) short introduction to latex, which will answer most of your questions (although typing your question into your favorite search engine will probably work well, too). You may also want to try https://cloud.sagemath.com/ or https://www.sharelatex.com/ for a cloud version.