AS/SC/MATH3410.03F
Complex Variables
Some polynomials, such as,
have no roots if
we confine ourselves to the real number system. The complex numbers
can be defined as the set of all numbers of the form
,
where and b are real,
i is a new kind of number satisfying
, and the operations of arithmetic
are carried out in a
fairly obvious way. The complex numbers include the reals (case),
and the extended system has the desirable property that not only
but every polynomial now
has a root. In the
system of complex numbers certain connections are seen between otherwise
apparently unconnected real numbers. A striking example is Euler's
formula. This is actually a
very simple consequence of the
extension to complex variables of the familiar exponential and
trigonometric functions. The concepts and operations of calculus
(differentiation, integration, power series, etc.) find their
most natural setting in complex (rather than real) variables. In
addition, some physical problems such as those involving
electrical circuits and certain twodimensional potential
problems (arising in fluid dynamics, airfoil theory,
electrostatics, etc.) are most easily analysed in the context of
complex numbers and functions.
The present course is intended to give the student a basic
knowledge of complex numbers and functions and a basic facility
in their use. The subject is a vast one, however, and its study
can be continued in MATH4210.03 (Complex Analysis).
Topics include: Complex numbers and their representations;
functions of a complex variable; mapping of elementary functions;
complex differentiation; CauchyRiemann equations, conformal
mapping and application to physical problems; complex
integration; Cauchy's theorem; Cauchy's integral formula and its
applications; complex power series; the residue theorem and its
applications.
A possible text is Richard A. Silverman, Complex Analysis with
Applications (Dover).
The final grade may be based on assignments, a midterm
examination and a final examination.

Prerequisites: AS/SC/MATH2015.03 or AS/SC/MATH3010.03 or
permission of the Course Coordinator.

Exclusions: MATH4210.06, ACMS3040.06.

Coordinator: R. G. Burns