AS/SC/AK/MATH 3410 3.0 W

Complex Variables

      Analytic functions, the Cauchy-Riemann equations, complex integrals, the Cauchy integral theorem, maximum modulus theorem. Calculations of residues and applications to definite integrals, two-dimensional potential problems and conformal mappings.
      Some polynomials, such as x²+ 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 i²= -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 x² + 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%).