# Math 332-002: Introduction to Functions of a Complex VariableSpring 2019 Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

## Course Information

Course Description: Functions of a complex variable: Cauchy-Riemann equations, Cauchy-Goursat theorem, integration, series, residues, poles, geometrical aspects. Emphasis on techniques. Effective From: Fall 2010.

Number of Credits: 3

Prerequisites: Math 211 or Math 213 and Math 222 all with a grade of C or better.

Course-Section and Instructors

Course-Section Instructor
Math 332-002 Professor D. Blackmore

Office Hours for All Math Instructors: Spring 2019 Office Hours and Emails

Required Textbook:

 Title Complex Variables and Applications Author Brown Edition 9th Publisher McGraw-Hill ISBN # 978-0073383170 Website http://web.njit.edu/~blackmor/

University-wide Withdrawal Date: The last day to withdraw with a W is Monday, April 8, 2019. It will be strictly enforced.

## Course Goals

Course Objectives:

• Gain deep understanding of the relevance and broad importance of the theory of analytic functions.
• Learn the meaning of theorems describing important properties of analytic functions, and understand their corollaries.
• Learn the deep connection between the series representations and integration properties of analytic functions.
• Learn applications of the Cauchy Residue Theorem, in particular its use in calculating certain definite integrals.
• Learn how to apply the knowledge of analytic functions to problems in applied mathematics, science and engineering.

### Course Outcomes

• Students gain deeper knowledge of the theory of analytic functions of a complex variable, and its broad applicability.
• Students gain deeper understanding of common elementary transcendental functions through the knowledge of their properties in the complex plane.
• Students are prepared for further study in more advanced mathematics, science and engineering courses.
• Students can apply their knowledge of the theory of analytic functions to solve problems in applied mathematics, fluid dynamics, electrodynamics, and other areas of science and engineering.

Course Assessment: The assessment of objectives is achieved through homework assignments and quizzes, and the in-class midterm and final examinations.

## Policies

DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

 Homework and Quizzes 20% Attendance 2% Midterm Exam 36% Final Exam 42%

Your final letter grade will be based on the following tentative curve.

 A 87 - 100 C 62 - 67 B+ 81 - 86 D 55 - 61 B 75 - 80 F 0 - 54 C+ 68 - 74

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced.

Homework and Quiz Policy: Homework problem sets will be emailed at the end of each week, and will be based on the material covered that week. Late homework will not be accepted. A short quiz based on the homework problems will be given about every other week, and will be announced at least one day in advance.

Exams: There will be one midterm exam held in class during the semester and one comprehensive final exam. Exams are held on the following days:

 Midterm Exam March 15, 2019 Final Exam Period May 10 - 16, 2019

The final exam will test your knowledge of all the course material taught in the entire course. Make sure you read and fully understand the Math Department's Examination Policy. This policy will be strictly enforced.

Makeup Exam Policy: There will be No make-up QUIZZES OR EXAMS during the semester. In the event an exam is not taken under rare circumstances where the student has a legitimate reason for missing the exam, the student should contact the Dean of Students office and present written verifiable proof of the reason for missing the exam, e.g., a doctor’s note, police report, court notice, etc. clearly stating the date AND time of the mitigating problem. The student must also notify the Math Department Office/Instructor that the exam will be missed.

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

## Additional Resources

Math Tutoring Center: Located in Cullimore, Room 214 (See: Spring 2019 Hours)

Further Assistance: For further questions, students should contact their instructor. All instructors have regular office hours during the week. These office hours are listed on the Math Department's webpage for Instructor Office Hours and Emails.

All students must familiarize themselves with and adhere to the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. The Department of Mathematical Sciences takes these policies very seriously and enforces them strictly.

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at lyles@njit.edu. The office is located in Fenster Hall Room 260. A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Spring 2019 Academic Calendar, Registrar)

Date Day Event
January 22, 2019 T First Day of Classes
February 1, 2019 F Last Day to Add/Drop Classes
March 17 - 24, 2019 Su - Su Spring Recess - No Classes, NJIT Open
April 8, 2019 M Last Day to Withdraw
April 19, 2019 F Good Friday - No Classes, NJIT Closed
May 7, 2019 T Friday Classes Meet/ Last Day of Classes
May 8 & 9, 2019 W & R Reading Days
May 10 - 16, 2019 F - R Final Exam Period

# Course Outline

Date Lecture Sections Topic
January 22 1 1-.5 Complex Algebra; Vectors & Moduli; Complex Conjugate
January 25 2 1-5 Polar Representation; Products & Powers in Exponential Form; Roots
January 29 3 12 Regions in the Complex Plane
February 1 4 13-14 Functions of Complex Variable; Mappings
February 5 5 15-18 Limits and Continuity
February 8 6 19-23 Derivatives & Analyticity; The Cauchy-Riemann Equations
February 12 7 24-26 Analyticity; Cauchy-Riemann Equations in Polar Coordinates
February 15 8 27-29 Harmonic Functions; Uniquely Determined Functions; Reflection Principle
February 19 9 30-36 The Exponential and Logarithm, The Power Function
February 22 10 37-39 Trigonometric and Hyperbolic Functions
February 26 11 40 Inverse Trigonometric & Inverse Hyperbolic Functions
March 1 12 41-49 Contour Integrals; Fundamental Theorem of Calculus
March 5 13 50-54 The Cauchy--Goursat Theorem & The Cauchy Integral Formula
March 8 14 55-59 The Extensions of the Cauchy Integral Formula
March 12 15 REVIEW FOR MIDTERM EXAM (Covers Lectures 1-13)
March 15 16 MIDTERM EXAM (Covers Lectures 1-13)
March 17-24 --- --- SPRING RECESS
March 26 17 60-65 Taylor Series; Power Series Convergence
March 29 18 66-68 Laurent Series
April 2 19 69-72 Uniform Convergence; Integration & Differentiation of Power Series
April 5 20 73-74 Series Multiplication, Division,  Composition
April 9 21 74-76 Cauchy's Residue Theorem
April 12 22 77-84 Zeros, Singularities & The Point at Infinity
April 16 23   85-87 Improper Integrals from Fourier Analysis
April 19  GOOD FRIDAY
April 23 24 88 Improper Integrals Continued: Jordan’s Lemma
April 26 25 89-90 Integrals Involving Indented Contours
April 30 26 91 Integration along a Branch Cut
May 3 27 92 Definite Integrals Involving Sines and Cosines
May 7 28 (Friday Classes Meet) REVIEW FOR FINAL EXAM

Updated by Professor D. Blackmore - 1/21/2019
Department of Mathematical Sciences Course Syllabus, Spring 2019