A complex function is a mathematical function that takes complex numbers as inputs and produces complex numbers as outputs. Complex numbers are numbers that have a real part and an imaginary part, typically expressed 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. These functions can be analyzed using various mathematical tools, such as limits, derivatives, and integrals, similar to real-valued functions. One important aspect of complex functions is their behavior in the complex plane, where they can exhibit unique properties like holomorphicity, which indicates that they are differentiable at every point in a given region.