Functions of a complex variable

Functions of a complex variable are mathematical rules that relate one complex number to another, much like a function in real numbers but extended to include imaginary parts. They map inputs consisting of a real and imaginary component to outputs that also have real and imaginary parts. These functions are fundamental in analyzing phenomena like fluid flow, electromagnetic fields, and signal processing, providing a powerful way to understand and solve problems involving two-dimensional quantities, often revealing deeper structural properties and behaviors when numbers are treated as points in a complex plane.