COLLEGE OF ARTS AND SCIENCES
Department of Mathematics and Physics
§ I: MATH 4423 Complex Variables Syllabus
Catalog Description
Review of elementary functions and Euler forms, holomorphic functions, Laurent series, singularities, calculus
of residues, contour integration, maximum modulus theorem, bilinear and inverse transformation, conformal
mapping and analytic continuation.
Required Textbook
A First Course in Complex Variables, Dennis G. Zill and Patrick D. Shanahan. Jones and Bartlett Learning, 3e,
ISBN 9781449694616 (2009). All students are encouraged to become familiar with the use of a variety of
software tools that augment the teaching of this course.
Course Objectives
The course, MATH 4423, provides students of mathematics and engineering with knowledge and prociency
in the theory and applications of complex analysis to the evaluation of certain real integrals, and solution of
boundary value problems. The emphasis is on connecting mathematical concepts, including complex integra-
tion, complex functions, residue theorem, conformal mappings, with some applications to boundary value
problems.
Student Learning Outcomes
After successfully completing this course the student:
1. Is able to perform algebraic operations with complex numbers, complex dierentiation and integration
using complex function theory.
2. Has an understanding of the main concepts of analyticity, residue theorem, conformal mapping, and
multi-valued functions.
3. Is able to apply the concepts of complex analysis to evaluate real integrals and boundary value prob-
lems.
Required Curriculum Content
Key topics covered include:
1. Introduction to complex numbers and the complex plane.
2. Complex functions, their properties and analytic properties in regard to calculus in the complex plane.
3. Series and residue integration.
The MATH 4423 Complex Variables will cover, as a minimum, the material from A First Course in Complex
Variables, Dennis G. Zill and Patrick D. Shanahan. Jones and Bartlett Learning, 3e, ISBN 9781449694616 (2009),
as listed:
Sec Textbook Topic
Chapter 1 Complex numbers and the complex plane
1.1 Complex numbers and their properties
1.2 Complex plane
1.3 Polar form and complex numbers
1.4 Powers and roots
1
Sec Textbook Topic
1.5 Sets of points in the complex plane
1.6 Applications
Chapter 2 Complex functions and mappings
2.1 Complex functions
2.2 Complex functions as mappings
2.3 Linear mappings
2.4 Special power functions
2.4.1 The power function z
n
2.4.2 The power function z
1/n
2.5 Reciprocal functions
2.6 Applications
Chapter 3 Analytic functions
3.1 Limits and continuity
3.1.1 Limits
3.1.2 Continuity
3.2 Dierentiability and analyticity
3.3 Cauchy-Riemann equations
3.4 Harmonic functions
3.5 Applications
Chapter 4 Elementary functions
4.1 Exponential and logarithmic functions
4.1.1 Complex exponential function
4.1.2 Complex logarithmic function
4.2 Complex powers
4.3 Trigonometric and hyperbolic functions
4.3.1 Complex trigonometric functions
4.3.2 Complex hyperbolic functions
4.4 Inverse trigonometric and hyperbolic functions
4.5 Applications
Chapter 5 Integration in the complex plane
5.1 Real integrals
5.2 Complex integrals
5.3 Cauchy-Goursat theorem
5.4 Independence of path
5.5 Cauchys integral formulas and their consequences
5.5.1 Cauchys two integral formulas
5.5.2 Some consequences of the integral formula
Chapter 6 Series and Residues
6.1 Sequences and series
6.2 Taylor series
6.3 Laurent series
6.4 Zeros and poles
6.5 Residues and Residue theorem
6.6 Some consequences of the Residue theorem
6.6.1 Evaluation of real trigonometric integrals
6.6.2 Evaluation of real improper integrals
Department Syllabus for MATH 4423, Spring 2019 Page 2 of 7 Rev. 1.0, January 28, 2019
Common Department Requirements for MATH 4423
While students in each section of MATH 4423 are assessed by the course instructor, there are general guide-
lines that apply to all sections of MATH 4423. These include:
Calculators and other electronic devices are not allowed on any exams.
Department Syllabus for MATH 4423, Spring 2019 Page 3 of 7 Rev. 1.0, January 28, 2019
Department, College and University Expectations and Policies
It is important that students familiarize themselves with a range of policies and guidelines that have been es-
tablished by the Department of Mathematics and Physics, the College of Arts and Sciences, and the University
of New Haven. These are an integral part of the syllabus for this course.
Adding/Dropping a Class
The nal day to drop this course without it appearing on your transcript is discussed on the
Academic Schedules and Registration web page. After the rst week of class, self-service registration will
not be enabled for students to directly add or drop classes. Students should contact the Registrars oce
directly or the Academic Success Center for assistance with adding and dropping courses during this time.
Attendance Regulations
University attendance policy guidelines require that:
Students are expected to attend regularly and promptly all their classes, appointments, and exercises. While the
university recognizes that some absences may occasionally be necessary, these should be held to a minimum.
A maximum of two weeks of absences will be permitted for illness and emergencies. The instructor has the
right to dismiss from class any student who has been absent more than the maximum allowed. A dismissed
student will receive a withdrawal (W) from the course if they are still eligible for a withdrawal per the university
Withdrawal from a Course policy, or a failure (F), if not. A student who is not ocially registered in the course
is not permitted to attend classes or take part in any other course activities. Students absent from any class
meeting are responsible for making up missed assignments and examinations at the discretion of the instructor.
Students are to adhere to the policy attendance policy guidelines outlined in the University Catalog under the
heading, Attendance Regulations, found online in the Undergradaduate Catalog or alternatively found in the
Student Handbook on pp.4849.
Religious Observance Policy for Students
The University of New Haven respects the right of its students to observe religious holidays that may neces-
sitate their absence from class or from other required university-sponsored activities. Students who wish
to observe such holidays should not be penalized for their absence, although in academic courses they are
responsible for making up missed work. The College provides that,
Instructors should try to avoid scheduling exams or quizzes on religious holidays, but where such conicts occur
should provide reasonable accommodations for missed assignment deadlines or exams. If a class, an assign-
ment due date, or exam interferes with the observance of such a religious holiday, it is the students responsibil-
ity to notify their instructor, preferably at the beginning of the term, but otherwise at least two weeks before the
holiday.
More information about religious observance policies can be found in the Student Handbook on pp.4849
under the heading, Attendance Policies: Religious Observance Policy for Students.
Withdrawal from a Course
Students wishing to withdraw must submit a request for an ocial course withdrawal in writing using the on-
line Course Withdrawal Form, or alternatively complete and hand in the pdf based Course Withdrawal Form.
The nal date to request a withdrawal is listed in the Academic Calendar. This request must be submitted to
the Registrars Oce and signed by the International Oce if you are an international student. The grade of
W will be recorded, but the course will not aect the GPA.
1
Incomplete Grade Policy
A grade of Incomplete (INC) is given only in special circumstances and indicates that the student has been
given permission by the instructor to complete required course work (with the same instructor) after the end
1
Please note that it is the responsibility of the student to assure that the required paperwork and documentation is completed by the deadline.
Department Syllabus for MATH 4423, Spring 2019 Page 4 of 7 Rev. 1.0, January 28, 2019
of the term. In the absence of the instructor a student should contact the Department Chair. Students need
to examine carefully the changed guidelines pertaining to INC grades, specically:
To remove the INC grade, the student must complete all required course work in timely fashion as stipulated by
the instructor but no later than the end of the following term. Fall and intersession course incomplete grades
must be completed no later than the last day of the spring term. Spring and summer course incomplete grades
must be completed no later than the last day of the fall term.
If the course work is not submitted within the allotted time, the INC grade will be changed to an F shortly after
the deadline by the Oce of the University Registrar. Students will be notied via campus email at least two
weeks prior to the change of grade process.
The University policy on incomplete grades is discussed in the Academic Catalog under the heading, Incom-
plete (INC) Grade Policy.
Academic Integrity Policy and Procedures
The University of New Haven expects its students to maintain the highest standards of academic conduct.
Academic dishonesty is not tolerated at the University. To know what it is expected, students are responsible
for reading and understanding the statement regarding academic honesty in the Student Handbook. Specif-
ically, students are required to adhere to the Academic Integrity Policies specied in the Student Handbook,
i.e., on pp.6673.
Please ask your instructor about their expectations regarding permissible or encouraged forms of student
collaboration if there is any confusion about this topic. The Department of Mathematics and Physics fully
adheres to the Academic Integrity Policy:
Academic integrity is a core university value that ensures respect for the academic reputation of the University,
its students, faculty and sta, and the degrees it confers. The University expects that students will conduct
themselves in an honest and ethical manner and respect the intellectual work of others. Please be familiar with
the Universitys policy on Academic Integrity. Please ask about expectations regarding permissible or encouraged
forms of student collaboration if they are unclear.
Coursework Expectations
This course will require signicant in-class and out-of-class commitment from each student. The University
estimates that a student should expect to spend two hours outside of class for each hour they are in a
class. For example, a three credit course would average six [6] hours of additional work outside of class.
2
Coursework expectations are detailed in the Academic Catalog under the heading, Course Work Expectations.
Please note, that MATH 4423 is a 3-credit course, and as such requires a total of 9 hours per week invested
in study and homework for the average student.
Commitment to Positive Learning Environment
The University adheres to the philosophy that all community members should enjoy an environment free of
any form of harassment, sexual misconduct, discrimination, or intimate partner violence. If you have been
the victim of sexual misconduct we encourage you to report this. If you report this to a faculty/sta member,
they must notify our colleges Title IX coordinator about the basic facts of the incident (you may choose to
request condentiality from the University). If you encounter sexual harassment, sexual misconduct, sexual
assault, or discrimination based on race, color, religion, age, national origin, ancestry, sex, sexual orientation,
gender identity, or disability please contact the Title IX Coordinator, Caroline Koziatek at (203)-932-7479 or
CKoziatek@newhaven.edu. Further online information about is available at Title IX.
Reporting Bias Incidents
At the University of New Haven, there is an expectation that all community members are committed to cre-
ating and supporting a climate which promotes civility, mutual respect, and open-mindedness. There also
exists an understanding that with the freedom of expression comes the responsibility to support community
2
Please note that study guidelines are important, i.e., there is substantial evidence that shows that the pass rates for students in math courses decrease
dramatically as the time spent on outside study falls below 2 hours of homework per credit per week.
Department Syllabus for MATH 4423, Spring 2019 Page 5 of 7 Rev. 1.0, January 28, 2019
members right to live and work in an environment free from harassment and fear. It is expected that all mem-
bers of the University community will engage in anti-bias behavior and refrain from actions that intimidate,
humiliate, or demean persons or groups or that undermine their security or self-esteem.
If you have witnessed or are the target of a bias-motivated incident, please contact the Oce of the Dean
of Students at 203-932-7432 or Campus Police at 203-932-7014. Further information about this and other
reporting options may be found at Report It.
University Support Services
The University recognizes students often can use some help outside of class and oers academic assistance
through several oces. In addition to discussing any academic issues you may have with your instructor,
advisor, or with the the courses or department coordinator or chair, the University provides these additional
resources for students:
The Center for Academic Success and Advising (CASA)
The Academic Success Center is located in Maxcy 208 for help with your academic studies, or call 203-932-
7234 to set up an appointment.
University Writing Center
The mission of the Writing Center (an expansion of the Writer to Writer peer-tutoring program) is to provide
high-quality tutoring to undergraduate and graduate students as they write for a wide range of purposes and
audiences. Tutors are undergraduate and graduate students and they work with students at any stage in the
writing process; Bring in your assignment, your ideas, and any writing done so far. To make an appointment,
register for an account at https://newhaven.mywconline.com.
The Math Zone
Please contact the Math Zone if you wish to challenge your Math Placement by taking a Math Challenge Exam
or by taking a Math Post Placement Exam. These are discussed more extensively at http://math.newhaven.
edu/mathphysics/placement_html. The Math Zone also provides a range of tutoring and classroom support
service for students taking development math classes.
The Center for Learning Resources (CLR)
The Center for Learning Resources located in Peterson Library, provides academic content support to the
students of the University of New Haven using metacognitive strategies that help students become aware
of and learn to apply optimal learning processes in the pursuit of creating independent learners CLR tutors
focus sessions on discussions of concepts and processes and typically use external examples to help students
grasp and apply the material.
Accessibility Resources Center
Students with disabilities are encouraged to share, in condence, information about needed specic course
accommodations. The Accessibility Resources Center (ARC) provides comprehensive services and sup-
port that serve to promote educational equity and ensure that students are able to participate in the oppor-
tunities available at the University of New Haven. Accommodations cannot be made without written docu-
mentation from the ARC. The ARC is located on the ground oor in the rear of Sheeld Hall. Sheeld Hall
is located in the Residential Quad area, and can be contacted at 203-932-7332. The ADA/Section 504 Com-
pliance Ocer is Rebecca Johnson, RJohnson@newhaven.edu, and can be reached by phone at 203-932-7238.
Information on the ARC can be found at
Department Syllabus for MATH 4423, Spring 2019 Page 6 of 7 Rev. 1.0, January 28, 2019
Counseling and Psychological Services
The Counseling Center oers a variety of services aimed at helping students resolve personal diculties and
acquire the balance, skills, and knowledge that will enable them to take full advantage of their experience at
the University of New Haven. Information about the, Counseling and Psychological Services, is available
online.
Department Syllabus for MATH 4423, Spring 2019 Page 7 of 7 Rev. 1.0, January 28, 2019