Complex variables are mathematical entities that extend the concept of real numbers to include imaginary numbers. A complex variable is typically expressed in the form z = x + iy, where x is the real part, y is the imaginary part, and i is the imaginary unit, defined as the square root of -1. This framework allows for the analysis of functions that depend on complex numbers, leading to rich mathematical structures and applications. In the study of complex variables, one important area is complex analysis, which focuses on functions of complex numbers and their properties. These functions can exhibit unique behaviors, such as holomorphicity, where they are differentiable in a complex sense. Complex variables are widely used in various fields, including engineering, physics, and applied mathematics, due to their ability to simplify calculations and model complex phenomena.