MTH 355 - CUE: Complex Analysis

Course unit(s): 1
Prerequisite(s): MTH 223 Calculus III and MTH 240 Transition to Abstract Mathematics
Throughout their mathematical education, students are taught that they can’t take the square root of a negative number, as this would lead to a non-real “imaginary” number. However, one can theoretically build the square roots of negative numbers into a number system called the complex numbers. Studying the properties of complex numbers and functions of a complex variable leads to many mathematically beautiful results which are arguably more elegant and natural than their real counterparts. Moreover, the results of complex analysis can be applied not only to other mathematical fields such as Number Theory and Differential Equations, but also to practical problems in engineering and the physical sciences. In this course, we will explore the algebraic properties of complex numbers, extensions of exponential and trigonometric functions to a complex variable, differentiable functions of a complex variable and their properties, complex line integrals, and some classical results such as the Maximum-Modulus principle, Taylor and Laurent series, Residue Calculus, and the Fundamental Theorem of Algebra.