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In mathematics, 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. They were discovered by , who later became the French prime minister. ==History== Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are poles. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the Painlevé property can be transformed into the Weierstrass equation or the Riccati equation, which can all be solved explicitly in terms of integration and previously known special functions. Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and tried and failed to find new examples with the Painlevé property. (For orders greater than 2 the solutions can have moving natural boundaries.) Around 1900, Paul Painlevé studied second order differential equations with no movable singularities. He found that up to certain transformations, every such equation of the form : (with ''R'' a rational function) can be put into one of fifty ''canonical forms'' (listed in ). found that forty-four of the fifty equations are reducible in the sense that they can be solved in terms of previously known functions, leaving just six equations requiring the introduction of new special functions to solve them. (There were some computational errors in his work, which were fixed by B. Gambier and R. Fuchs.) It was a controversial open problem for many years to show that these six equations really were irreducible for generic values of the parameters (they are sometimes reducible for special parameter values; see below), but this was finally proved by and . These six second order nonlinear differential equations are called the Painlevé equations and their solutions are called the Painlevé transcendents. The most general form of the sixth equation was missed by Painlevé, but was discovered in 1905 by Richard Fuchs (son of Lazarus Fuchs), as the differential equation satisfied by the singularity of a second order Fuchsian equation with 4 regular singular points on P1 under monodromy-preserving deformations. It was added to Painlevé's list by . tried to extend Painlevé's work to higher order equations, finding some third order equations with the Painlevé property. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Painlevé transcendents」の詳細全文を読む スポンサード リンク
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