1701.430 - Introduction to Complex
Analysis I
Winter/Spring, 2007
Dr. Marcus W. Wright – wright@rowan.edu
- 256-4500 3873
My office is Robinson 229D and the door is always open to you.
My tentative office hours are MW 2-3, TR and by appointment M - F. Please
feel free to stop by any time.
Prerequisite: Introduction to Real Analysis I and Calculus III
The purpose of this course is to provide an intuitive but also rigorous
introduction to the theory and applications of complex numbers and functions
of a single complex variable. This course is designed to be appropriate
for majors in mathematics, physics, and engineering. This course is
essential for serious students of both pure and applied mathematics,
especially those who are considering graduate training in mathematics
or related disciplines.
Required Texts:
1 Complex Variables: Second Edition. Stephen Fisher. Dover
Publications ISBN -486406702
2. Schaum”s Outline of Complex Variables, Murray Spiegel. McGraw-Hill
- It is very important for you to see the same topics handled by more
than one author. This helps you to become more flexible in learning mathematics..
Studying both works can help you get a thorough introduction to complex
analysis this semester.
We will also use: An Intuitive Introduction to Complex Variables,
by Thomas Osler, available from his Rowan website. This text is very effective
in conveying the facts about complex variable analysis. I may assign problems
from it.
GRADING:
3 tests will count for 60% (may have take home components...)
Homework will count for 25% It must be handed in when due. (Also, a homework
portfolio MUST be handed in at end of semester...)
A research project will be worth 15%
-- You can begin now to search for a topic for this project. I can give
you some ideas any time.
Overview of the Content of the Course
--Basic algebraic and geometric properties of the complex numbers
Polar form and De Moivre's Theorem. Roots of complex numbers. The
complex plane and its topology. The Fundamental Theorem of Algebra.
-- The basic "calculus" of functions of a complex variable
continuity, differentiation and integration.
There are many parallels with these concepts in real analysis, but there
are also many perhaps surprising and significant differences. The Cauchy
Theorem is very important.
-- The Idea of a Riemann surface
-- The Cauchy Integral Formula and some of its consequences:
-- Taylor and Laurent Series
-- The Residue Theorem
-- Mapping properties of complex functions
-- The gamma function and related
functions
NOTES:
Some of the things that make this such a fundamental subject
of mathematics are its connections to:
- real analysis of one variable
- topology and geometry of the Cartesian plane
- real analysis of two real functions of two real variables
- analytic geometry
- abstract algebra, which is able to enter because of very
deep reasons related,
for example to the Fundamental Theorem of Algebra, and to the surprising
strength of the hypothesis of differentiability for a complex function
- the results of the utter, absolute genius of the many of the great
mathematicians of the 18th, 19th, and 20th centuries
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