Pierre Fatou was the first to define the Mandelbrot set.
Fatou was particularly interested in the case where Z_0 = 0, which was later analysed with computers by Benoît Mandelbrot to generate graphical representations of the behaviour of this series for each point, c, in the complex plane - now popularly called the Mandelbrot set.
Fatou wrote many papers developing a Fundamental theory of iteration in 1917, which he published in the December 1917 part of Comptes Rendus. His findings were very similar to those of Gaston Maurice Julia, who submitted a paper to the Académie des Sciences in Paris for their 1918 Grand Prix on the subject of iteration from a global point of view. Their work is now commonly referred to as the Generalised Fatou Julia theorem.