If., in the vicinity of a singular point, the solutions are regular, they may be developed in the manner indicated. The problem of finding the developments of the solution in the vicinity of a point where they are not regular is far more difficult and still awaits a satisfactory general solution. A solution, not regular at x , may have the special form.
eUtx—a)PiPtx), where p is a constant. where 1P(x) is an ordinary power-series in x--a, and where a. as as — (x—a) a (x--a) a X—a so that it differs from a regular integral only by the presence of the factor ei2. Such an in tegral, if it existi, is called a normal integral. There may also be integrals of a similar form in which, however, (x—a) 1/0, appears in place of x —a, where k is a positive integer. They are called subnormal. The-conditions for the exist ence of normal and subnormal integrals have been investigated, but none of these investiga tions is as yet in a final form. Considerable progress in the theory of non-regular integrals has been made in recent years by Birkhoff.
It is possible however to change the method of attack. The general theory shows that, in the vicinity of the singular point a, a solution exists of the form (x—a)P0(x), where 0(x) is, in general, a Laurent series. The question is this: how to determine the exponent p and the coefficients of cp(x). In the regular case, when cb(x) is an ordinary power series, substitution of this expression into the differential equation, and comparison of powers of x —a, solves the problem. One may do the same thing in general. But then one finds it necessary to solve a system of linear equations infinite in number and with an infinity of unknown quantities. This leads to the notion of infinite determinants, due primarily to G. W. Hill. Hill applied infinite deter minants just as though they were finite, paying no attention to convergence or rigorous defi nitions. This deficiency was made up and t.he whole theory placed upon a solid basis by Poincare and Koch.
The theory of linear differential equations has served as a basis for practically all that is Icnown about non-linear equations. There are two fundamental properties of the linear equa tions which render them peculiarly accessible. In the first place it is known, a priori, how the arbitrary constants enter into the expression of its general integral; in the second place the singular points of its solutions are fixed, i.e. independent of the constants of integration.
Other classes of differential equations may be defined which have one or both of these proper ties. The first-mentioned point of view leads to the differential equations with fundamental solutions. These may be defined in various ways and have been investigated by Guldberg, Vessiot, Lie and Wilczynslci. The idea of in vestigating the differential equations with fixed branch-points is due to Fuchs. For equations of the first order he succeeded in formulating the conditions in a very simple theorem. Pain care then showed that all such equations can be transformed into a Ricati equation, i.e., an equation of the form dy (36) ao+ctoo, dx where a., a., al are functions of x, or else are integrable by quadratures or algebraic functions. Differential equations of the first order with fixed branch-points do not, therefore, as was at first expected, lead to new transcendental func tions. For, the Riccati equation may, by the 1 dt betransformation y= converted into a dx linear differential equation of the second order. It may be noted, incidentally, that this remark enables us to prove, in a simple manner, the theorem that the anharmonic ratio of any four solutions of a Riccati equation is constant. This is important in geometric applications.
The most important recent investigations in the theory of differential equations, from the standpoint of the theory of functions of a com plex variable, are due. to Pain!eve. A brief ac count of some of them will indicate their funda mental nature. Let dy (37) Y) be an algebraic differential equation of the first order. The general integral will be a function of x and u, u being the constant of integration. We may, instead, consider u as a function of x and y defined by the partial differential equation au au (3 8) — — f(x. y) =--- O.
ax ay The general integral of (37) is said to be reducible if other equations, algebraic in x, y, u, au au Mu etc., may be adjoined to (38) com ax ay axs patible with it without being deducible from. All of the equations of the first order which have been studied are reducible in this sense; for instance, the Riccati equation, the linear equation, etc. In the case of a linear altu equation the condition— or— =--0 may be thus ad joined; if the equation admits an algebraic integrating factor X, we may adjoin the condi tion X; etc. This definition of reducibility ay may be extended to equations, or systems of equations, of any order.