Functions of one Complex Variable

Complex variables is a beautiful area from a purely mathematical point of view, as well as a powerful tool for solving a wide array of applied problems. It is related to many mathematical disciplines, including in particular real analysis, differential equations, algebra and topology. The numerous applications include all kinds of wave propagation phenomena such as those occurring in electrodynamics, optics, fluid mechanics and quantum mechanics, diffusion problems such as heat and contaminant diffusion, engineering tasks such as the computation of buoyancy and resistance of wings, the flows in turbines and the design of optimal car bodies, and signal processing and communication theory.

Historically, complex numbers originated from the desire to find a uniform representation of solutions of algebraic equations. From this perspective, the field of complex numbers is a natural extension of the field of real numbers with the property that it is algebraically closed, that is, every polynomial can be factorized into linear polynomials. After analysis has been introduced, it became a natural task to extend the concepts of differential and integral calculus and function series to complex variables. In the 19th century complex analysis emerged as an independent mathematical discipline, most notably through the work of Augustin-Louis Cauchy (1789-1857), Karl Weierstrass (1815-1897), and Bernhard Riemann (1826-1866).

The purpose of this course is an introduction to the theory and application of complex variables and complex functions. Part I is devoted to the basic mathematical theory, and Part II to selected applications.

Contents:

The course covers the basic principles (both theory and applications) of differentiable complex-valued functions of a single complex variable. Topics include:

• the complex number system
•  Cauchy-Riemann conditions
•  analytic functions and their properties
•  special analytic functions including linear fractional transformations, roots, exponential, Log, trigonometric and hyperbolic functions of a complex variable
•  Complex integration and line integrals
•  Cauchy's theorem, Cauchy representation
•  conformal mapping
•  Taylor and Laurent Series expansions
• the calculus of residues and various applications
• graphical representations of analytic functions.

Learning Outcomes:

• Easily describe domains and compute limits in the complex plane.
• Verify analyticity of functions.
• Use line and contour integration to evaluate integrals.
• Calculate Taylor or Laurent series for functions.
• Verify convergence or divergence of complex series.
• Use residues to evaluate integrals.
• Compute Fourier and Laplace transforms.

Textbook:TBA