The proof of this theorem, due to Fuchs, is also based on the method of dominating func tions. The important point is the fact that the true radius of convergence of the series is determined by inspection from the differential equation itself. The existence of a fundamental system of solutions expressible by power-series follows at once.
Let yi, yn be the members of such a funda mental system. Let au . am be the singular points (poles) of the coefficients . . pn, which we shall assume to be rational functions of x. Let y,, ... yn be continued analytically along a path passing, in the positive direction, around one of these singular points a, and let . . . yn be the new branches of the functions yl, . . . yn which are thus defined by power-series in the vicinity of x=x0 after this process. We must have (28) 2, ...n), where aki are constants, since 5. . . . yn must constitute again a system of solutions (more over a fundamental system). A new funda mental system may be chosen in the following manner. Put co, + cos . . . cnyn, where c,, en are constant coefficients. After the continuation around a, s will be changed into l=c1(any, . aneyn) .
cn(anayi+ ... +annyw).
This will be equal to us, where 4) is a constant, if — Go) + clan . . • ± cnana ---- 0, cian-Fcl(an—W) + cnant-=-0, • • c.,&,,,,-61) =0, whence an—W, au, . • • ant (30) F(0 =_—_ ant an—to, a,„ =0.
If rr, is a root of (30) and the ratios of • cn are determined from (29) after w has been put equal to (di, we shall therefore find a solution si of (27) which changes into croft when the variable x describes a closed path around the singular point considered. If the equation F(4)).= 0 has n distinct roots, we shall find n such solutions, and we may write (31) (Jai, (i=1, 2, ... n) in place of (28). Moreover, these n solutions • . . . En will constitute a fundamental system.
We shall not attempt to discuss the case of co incident roots of the equation (30), which is known as the fundamental or characteristic equation.
Now the function (x—a)ri.-----eri 1ns (x-4), pi= 2 irs --,, log tri has precisely the same property. Therefore the quotient a function uniform in the vicinity of x'= a, and therefore expressible by a so-called Laurent series proceeding according to positive and negative but integral powers of x—a. Let 02(x) be such series; then we have (32) si=--(x—a)ro;02(x), (1--- 1, 2, . n).
The Laurent series will be convergent for all points, excepting a itself, of the circle which has a as center and which reaches up to the nearest singular point of the differential equation. The
main questions to solve are: 1st. Determine the exponents ri; 2d. Find the coefficients of the Laurent series tisi. These questions are capable of a direct and general solution in the special case in which the Laurent series contains only a finite number of terms involving negative powers of x—a. In that case the differential equation (27) may be written in the form.
day Pilx) dn—lyP2(x) dn—i X"--a dxn-2 kx--a)2dxn-2 Pn(x) —0, where Pi, PI, . . Pn are expressible as power series proceeding according to positive, integral powers of x—a. The exponents ri are then the roots of the determinating fundamental equation of the nth degree (34) r(r —1) ... (r—n + 1) +Pi(o)r(r — 1) .
(r —n ÷ 2) . . Pn(a) = O. After ri has been obtained from this equation, the method of indeterminate coefficients enables one to find the coefficients of the power-series. In the case of equal roots some of the solutions may contain such terms as log (x— a), log (x—a) 2, etc.• the general discussion of the various cases whi'ch may arise is rather com plicated.
The case in which the equation may be written in the form (33) is usually described as that in whid2 the solutions are regu!ar about x -------a. If they are regular in the vicinity of each singular point, including x= , the equa tion is said to be of the Fuchsian type, and may be written as follows: GP-t (35) y00+ —yOs-0 Crn(p—i) y =-- 0, where y", etc., denote the derivatives of y of the first, second order, etc, where (35a) 1P-=(x—a,)(x—a-i) . • (x--am), a,, . • . am and so being the singular points, and where GA denotes a poly-nominal in x of degree no higher than A. The most important special case of such an equation is that of the hyper geometric series, the so-called Gauss equation, which is of the second order and has three singular points, 0, 1 and co. Historically, the theory of the Gauss equation, as treated by Riemann, was the origin of the general theory of linear differential equations. A large num ber of the most important conceptions of the theory of functions are closely connected with this equation. The question of finding the cases in which the general solution is algebraic led Schwarz, Fuchs and Klein to the remark able algebraic functions which are connected with the five regular solids. This equation also leads to the general theory of automorphic functions, of which the elliptic functions are a special case.