Analytic functions, the
CauchyRiemann equations, complex integrals, the Cauchy integral
theorem, maximum modulus theorem. Calculations of residues and
applications to definite integrals, twodimensional potential
problems and conformal mappings.
Some polynomials, such as x^{2}+ 1 ,
have no roots if we confine ourselves to
the real number system, but do have roots if we extend the number system
to the complex numbers, which can be defined as the set of all numbers of the
form a + ib , where a and b are real and
i is a new kind of number satisfying
i^{2}= 1, where the basic arithmetic operations have the same structure as
those of the real numbers. The complex numbers include the reals
(case b
= 0), and the extended system has the desirable property that not
only x^{2}
+ 1 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 e^{i p} + 1 = 0 ,
which is a
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. The present course
is intended to give the student a basic knowledge of complex numbers and
functions and a basic facility in their use.
Further topics include: Complex numbers and their representations;
functions of
a complex variable; extensions of elementary functions from real to complex
variables; mapping of elementary functions; complex differentiation;
Cauchy's theorem; Cauchy's
integral formula and its applications; complex power series; the residue
theorem and its applications.
The text has not been chosen. The grade will be determined by
term work (60%) and a final exam (40%).
Prerequisite:AS/SC/MATH 2010 3.0 or AS/SC/MATH 2015 3.0 or
AS/SC/AK/MATH 2310 3.0. (AS/SC/AK/MATH 3010 3.0
is also recommended
as a prerequisite for students who have taken AS/SC/MATH 2010
3.0.)
Coordinator: S. Scull.
