Pierre Fatou

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Mandelpart4.jpg
Pierre Fatou was the first to define the Mandelbrot set.

Pierre Joseph Louis Fatou (28 February 1878, Lorient - 10 August 1929, Pornichet) was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory. Fatou continued his mathematical explorations and studied iterative and recursive processes like

<math>Z \to Z^2 + c \,\!<math>, where Z is a number in the complex plane of the form <math>Z=x+iy \,\!<math>, resulting in a series
<math>Z_0 = x+iy \,\!<math>
<math>Z_1 = Z_0^2+c \,\!<math>
<math>Z_2 = Z_1^2+c=(Z_0^2+c)^2+c = Z_0^4+2cZ_0^2+c^2+c\,\!<math>
<math>Z_3 = ... \,\!<math>

Fatou was particularly interested in the case where <math>Z_0 = 0<math>, which was later analysed with computers by Benoit 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. If the series does not tend to infinity, it is in the Mandelbrot set otherwise it is not. Images of the set are typically coloured black where points are in the set and coloured according to the escape speed of the series where not.

Each point in the complex plane also has an associated Julia set. There is a 1-1 correspondence between the points in the Mandelbrot set and the connected Julia sets. All points not in the Mandelbrot set have Julia sets that are not connected.

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.

See also

Fatou's lemma, Fatou set

External link

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