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.