A complex function is a mathematical function that takes complex numbers as inputs and produces complex numbers as outputs. Complex numbers are composed of 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 calculus and algebra. One important aspect of complex functions is their ability to be represented graphically in a two-dimensional plane known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.