# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 185A: Complex Variables

Approved: 2003-10-01 (revised 2022-06-02, J. Hunter, J.Schultens, and B.Temple)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Complex Variables and Applications, Ninth Edition by James Brown and Ruel V. Churchill; McGraw Hill Publisher
Search by ISBN on Amazon: 978-0073383170
Prerequisites:
(MAT 067 or (MAT 022A or MAT 027A or BIS 027A, MAT 108)), MAT 127B
Suggested Schedule:
 Lecture(s) Sections Comments/Topics 1-3 Chapter 1 Complex number system 4 Chapter 2 (pages 37-43) Functions of a Complex Variable 5 Chapter 2 (pages 44-51) Limits, Theorems on Limits, Limits involving the Points at Infinity 6-7 Chapter 2 (pages 52-67) Differentiation, Cauchy-Riemann equations 8 Chapter 2 (pages 68-75) Polar coordinates, Analytic functions 9-11 Chapter 3 Elementary Functions 12-13 Chapter 4 (pages 115-134) Contour Integrals 14 Chapter 4 (pages 140-147) Antiderivatives 15 Chapter 4 (pages 148-153) Cauchy-Goursat Theorem 16-17 Chapter 4 (pages 154-171) Cauchy Integral Formula, Derivatives of Analytic Functions 18 Chapter 4 (pages 172-178) Liouville's Theorem, Maximum Modulus Theorem 19-20 Chapter 5 (pages 179-201) Taylor Series, Laurent Series 21 Chapter 5 (pages 208-220) Convergence of Power Series, Integration and Differentiation of Power Series, Uniqueness 22 Chapter 5 (pages 221-226) Multiplication and Division of Power Series, Analytic Continuation 23-24 Chapter 6 (pages 227-237) Residues, Residue Theorems 25-26 Chapter 6 (pages 238-258) Zeros, Poles of Order m, Removable Singular Points
Learning Goals:
This is an introductory course on the basic concepts and theory of functions of complex variables and their applications. Students will learn how to apply ideas and techniques from Calculus to formulate the definition of a complex analytic function and to study special properties of complex analytic functions. These functions give rise to a very rich and beautiful mathematical theory with numerous applications in mathematics and beyond. Complex analysis appears in many areas of the natural sciences and has wide applications. Students will be challenged to improve their problem solving skills within the broad context of Calculus. Mastery of this course also supports the development of clear analytical thinking.
Assessment:
To assess the learning outcome for this course, the course will be graded through regular homework, midterm examinations, and a comprehensive final exam.