Complex Variables

Some polynomials, such as$x^2, 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 $a, where$a and b are real, i is a new kind of number satisfying $i^2, and the operations of arithmetic are carried out in a fairly obvious way. The complex numbers include the reals (case$b), and the extended system has the desirable property that not only $x^2 but every polynomial now has a root. In the system of complex numbers certain connections are seen between otherwise apparently unconnected real numbers. A striking example is Euler's formula$e^i*pi. 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 two-dimensional 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; Cauchy-Riemann 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.