The Julia set is named after the French mathematician Gaston Julia who investigated their properties circa 1915 and culminated in his famous paper in 1918. While the Julia set is now associated with a simpler polynomial, Julia was interested in the iterative properties of a more general expression, namely z⁴ + z³/(z-1) + z²/(z³ + 4 z² + 5) + c.

The Julia set is now associated with those points z = x + iy on the complex plane for which the series z_n+1 = z_n² + c does not tend to infinity. c is a complex constant, one gets a different Julia set for each c. The initial value z₀ for the series is each point in the image plane. In the broader sense the exact form of the iterated function may be anything, the general form being z_n+1 = f(z_n), interesting sets arise with nonlinear functions f(z). Commonly used functions include the following: