PAINLEVÉ TRANSCENDENTALS

PAINLEVÉ, PAUL.

Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme.

Berlin, Stockholm, Paris, Beijer, 1902. 4to. Bound in contemporary half cloth with gilt lettering to spine. In "Acta Mathematica", Vol, 25, 1902. Entire volume offered. Stamps to title page, otherwise a fine and clean copy. pp. 1-86.[Entire volume: (4), 383 pp].

First appearance of Painlevé's important paper in which he introduced some of his transcendents, now know as "Painlevé transcendents".

Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions.

"In old problems in which the difficulties seemed insurmountable, Painlevé defined new transcendentals for singular points of differential equations of a higher order than the first. In particular he determined every equation of the second order and first degree whose critical points are fixed.
The results of these studies are applicable to the equations of analytical mechanics which admit rational or algebraic first integrals with respect to the velocities. Proving, in the words of Hadamard’s éloge, that "continuing [the work of] Henri Poincaré was not beyond human capacity," Painlevé extended the known results concerning the n-body problem. He also corrected certain accepted results in problems of friction and of the conditions of certain equilibriums when the force function does not pass through a maximum."

Paul Painlevé later became the French prime minister.

Order-nr.: 49643