Math 355-10          Functions of a Complex Variable          Fall 2007
Instructor	Office	Phone	Hours
Jimmie Lee Johnson	Auditorium Building Chicago, Illinois Room 416	(312) 341-3552 Voice Mail	Tu 4:30-6:00pm and by appointment.
6:00-8:30pm Aud 401			I am often around earlier.
	E-mail:  jjohnson@roosevelt.edu Web Page:  http://faculty.roosevelt.edu/jjohnson/

Text: Complex Variables and Applications, by James Ward Brown and Ruel V. Churchill, McGraw Hill., 7th ed. 2004..

Prerequisite: Grades of C or better in Math 233 Calculus III and Math 290 Mathematical Reasoning. Math 300 Linear Algebra is also recommended.

Class will consist of oral reports, quizzes, examples, theorem proving and the answering of questions. Prior to each class, you must do the assigned reading or you will be lost; make notes of topics you feel need elaboration in class. For each class, be prepared to give a short explanation of a definition or proof of a theorem orally, at the chalkboard. You are responsible for all assigned reading even if it is not discussed in class.  Class will usually include
     1. a short quiz each week;
     2. now and then, a short oral presentation of current material by selected students;
     3. discussion of new material via examples and the proof of new theorems.

Quizzes will be given each week beginning September 14th, except for the weeks of the exams. The quizzes will be closed book, and will count for 10% of your grade. The two lowest quiz grades will be dropped. No make-ups; a missed quiz will count as one of the dropped grades.

Homework will be collected, discussed, graded and returned; Homework, along with oral presentation scores, will count for 20% of your grade. Late homework may be downgraded, if it is later than a week, but it is still worth more than no credit at all. Use of Maple software is recommended.

Exams will be given on October 23rd and November 20th. These will be closed book exams. The average of the exam swill count for 35% of your grade. No make-ups except for excused absences with advance notice. The grade on the subsequent exam will be used for both.

Final Examination will be given on Tuesday, December 18th, closed book and comprehensive. It will count for 35% of your grade.

Grades: Regulations covering grades (especially I and W) are on pages 254-255 of the 2006-2008 Undergraduate Catalog or on pages 203-205 of the 2005-2007 Graduate Catalog. Incompletes will not be given, except to a student who has done passing work up to the Final Examination (including most of the homework) but misses the final exam because of an excused absence with advanced notice. The last day to drop a class (with a grade of "W") is Friday, November 16th, and the drop form must be submitted to the Registrar's Office. Anyone registered after that must be graded solely on academic performance.

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Objectives:  The student is expected to be proficient in complex arithmetic, the solution of equations requiring complex roots, the identification of continuous complex functions, the evaluation of complex derivatives and integrals, and the determination of complex infinite series. To this end, the completion of a variety of homework problems and the occasional oral presentation are expected of each student. For assessment purposes, copies of some graded homework assignments and exams of each student will be placed in a file in the school office. Further review of these materials for assessment purposes will not affect the student's standing.

Syllabus

Date Approx. Pages	Sections	Topics -  changeable
September 11   22 pages	1.1 1.2 1.3 1.4 1.5 1.6 1.7	Sums and Products Basic Algebraic Properties Further Properties Moduli Complex Conjugates Exponential Form Products and Quotients in Exponential Form
September 18   24 pages	1.8 1.9 1.10 2.11 2.12 2.13 2.14	Roots of Complex Numbers Examples Regions in the Complex Plane Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits
September 25   19 pages	2.15 2.16 2.17 2.18 2.19 2.20 2.21	Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability
October 2   25 pages	2.22 2.23 2.24 2.25 2.26 2.27 3.28	Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle The Exponential Function
October 9   21 pages	3.29 3.30 3.31 3.32 3.33 3.34 3.35	The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
October 16   19 pages	4.36 4.37 4.38 4.39 4.40	Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Examples ---Review
October 23   12 pages	Exam #1 4.41 4.42 4.43	Chapters 1, 2, 3 - Lecture afterward. Upper Bounds for Moduli of Contour Integrals Antiderivatives Examples
October 30   33 pages	4.44 4.45 4.46 4.47 4.48	Cauchy-Goursat Theorem Proof of the Theorem Simply and Multiply Connected Domains Cauchy Integral Formula Derivatives of Analytic Functions
November 6   20 pages	4.49 4.50 5.51 5.52 5.53 5.54 5.55 5.56	Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle Convergence of Sequences Convergence of Series Taylor Series Examples Laurent Series Examples
November 13   15 pages	5.57 5.58 5.59 5.60 5.61 6.62 6.63 6.64 6.65 6.66	Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representation Multiplication and Division of Power Series Residues Cauchy's Residue Theorem Using a Single Residue The Three Types of Isolated Singular Points Residues at Poles --- Review
November 20	Exam #2	Chapters 4, 5. - No Lecture afterward. Out of Town.
November 21-25	No Classes	Thanksgiving Break
November 27   37 pages	6.67 6.68 6.69 6.70 7.71 7.72 7.73	Examples Zeros of Analytic Functions Zeros and Poles Behavior of  f  Near Isolated Singular Points Evaluation of Improper Integrals Examples Improper Integrals from Fourier Analysis
December 4   23 pages	7.74 7.75 7.76 7.77 7.78 7.79 7.80 8.83 8.84 8.85 8.86 8.87 8.88	Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane
December 11   14 pages	8.89 8.92 9.94 9.95	The Transformation w = sin z Riemann Surfaces Preservation of Angles Scale Factors --- Review
December 18	Final	Comprehensive.

This page is at http://faculty.roosevelt.edu/jjohnson/Fall2007/M355F07.htm and was last revised December 4, 2007.