The course starts by familiarizing students with the algebra of complex numbers.
Attention is given to some elementary topological concepts such as open sets of the plane.
The central concept of an analytic function is defined, and the familiar exponential, logarithmic, and trigonometric functions are extended into the complex domain. The Cauchy-Riemann equations arise naturally as a constraint imposed by analyticity on the real and imaginary parts of a function. Harmonic functions are discussed in this context as well. At its core, the course examines the theory of contour integrals, important because of the frequency with which they arise in various applications. We also review the basic notions of convergence of series and apply them to the formulation of analytic functions as convergent Taylor series in disks.