Complex functions are mathematical expressions that involve complex numbers, which are numbers that have a real part and an imaginary part. A complex number is typically written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as the square root of -1. Complex functions can be represented as f(z) = u(x, y) + iv(x, y), where z is a complex variable, and u and v are real-valued functions of the real variables x and y. These functions are important in various fields, including engineering, physics, and applied mathematics. They can be analyzed using techniques from calculus, such as differentiation and integration, but with specific rules that apply to complex numbers. One key concept is analyticity, which describes functions that are differentiable in a neighborhood of every point in their domain. This