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 DIFFERENTIAL divided respectively by 1, 1!, 2!, &c., then the differential equations determine uniquely all the other coefficients, and that the resulting series are convergent. We rely, in fact, upon the theory of monogenic analytical functions (see art. Functions, Analytic), a function being determined entirely by its development in the neighbourhood of one set of values of the independent variables, from which all its other values arise by continuation; it being of course understood that (the coefficients in the differential equations are to be continued at the same time. But it is to be remarked that there is no ground for believing, if this method of continuation be utilized, that the function is single-valued ; we may quite well return to the same values the independent variables aswith different Singular ya ^ue 0of f the function, belonging, we asay, to a points of q[jferent branch of the function ; and there is even no solutions. reason for assuming that the number of branches is finite, or that different branches have the same singular points and regions of existence. Moreover, and this is the most difficult consideration of all, all these circumstances may be dependent upon the values supposed given to the arbitrary constants of the integral; in other words, the singular points may be either fixed, beintf determined by the differential equations themselves, or they may°be movable with the variation of the arbitrary constants of integration. Such difficulties arise even in establishing the rever/ (j[% 2 sion of an elliptic integral, in solving the equation ( — J -(x- oq) (x - a2)(x - a3)(x - a4); about an ordinary value the right side is developable ; if we put x-a1 = tfi, the right side becomes developable about ^ = 0 ; if we put x—t, the right side of the changed equation is developable about 2 = 0 ; it is quite easy to show that the integral reducing to a definite value x0 for a value s0 is obtainable by a series in integral powers; this, however, must be supplemented by showing that for no values of s does the value of x become entirely undetermined. These remarks will show the place of the theory now to be sketched of a particular class of ordinary linear homogeneous differential equations whose importance arises from Linear the completeness and generality with which they can differen- be discussed. We have seen that if in the equations tial equa- dyldx=y 1ldx=y2, • • • ,dyn_fidx=yn_x,dyn_fidx= tions with a y + a _L,ydy n n x x+ • • +axyn^ wherein ax, a2, • • •, an are rational co "now taken to be rational functions of x, the value efficients. x = x°tobebeone for which no one of these rational functions is infinite, and y°, yx°, • • •, y°n-x be quite arbitrary finite values, then the equations are satisfied by y — y°u + yl°u1 + • ■ + y0n-iun-i, wherein u, ux, ■ • un_x are functions of x, independent of y° ■ ■ y°n-, developable about x=x°; this value of y is such that for 0x=x° the functions y, y1- ■ yn- reduce respectively to y°yfi • • V n~; it can be proved that the region of existence of these series extends within a circle centre a:0 and radius equal to the distance from a;0 of the nearest point at which one0 of eq • • an becomes infinite. Now consider a region enclosing a:, and only one of the places, say S, at which one of cq • • an becomes infinite. When x is made to describe a closed curve in this region, including this point 2 in its interior, it may well happen that the continuations of the functions u, ux, • • •, un_1 give, when we have returned to the point x, values v, vx, • •, vn^x, so that the integral under consideration becomes changed to y°v + yfiv^ + • • + At xT let this branch and the corresponding values of yx ■ ■ yn- be YYi • • y°n—i ; then, as there is only one series satisfying the equation and reducing to (r)0y0x • • y°n-i) for x = x°, and the coefficients in the differential equation are single-valued functions, we must have rj°u + H 1- 'non-1un_1=y0v + y°lv1 -1 1- y0n-iVn-i; as this holds for arbitrary values of y° • • y°n-i> upon which u ■ •«„_! and v • • Vn^ do not depend, it follows that each of v • • r„_1 is a linear function of u • • ■Mn_1 with constant coefficients, sayq=AijM+ • • +Ainun_1. Then y°v + • • + y0n-iVn-i = (fiiA^yi0) u+- • 0+ (2jAi„y0i)w„_1; this is equal 0to y(y°u+ • • +y0n-iUn-i) if 'Zikiry i=p.y°r_x ; eliminating y° • • y n- from these linear equations, we have a determinantal equation of order n0 for y; let yx be one of its roots ; determining the ratios of y yx° • • y°n-% to satisfy the linear equations, we have thus proved that there exists an integral, H, of the equation, which when continued round the point 2 and back to the starting-point, becomes changed to HjrzjiqH. Let now £ be the value of x at 2, and rx one of the values of log yx ; consider the function (ar —£)_riH; when x makes_ra circuit round a; = £, this becomes changed to exp{ - ‘lirir x) {x - £) i/i1H, that is, is unchanged; thus we may put H = (x - £)ri01, 0i being a function single-valued for paths in the region considered described about 2, and therefore, by Laurent’s Theorem (see art. Functions, Analytic), capable of expression in the annular region about this point by a series of positive and negative integral powers of x —£, which in general may contain an infinite number of negative powers ; there is, however, no reason to suppose rx to be an integer, or even real.

455

EQUATIONS

Thus, if all the roots of the determinantal equation in y are different, we obtain n integrals of the forms (x - £)r101, • • •, r (x-£) n0n. In general we obtain as many integrals of this form as there are really different roots ; and the problem arises todiscover, in case a root be 1c times repeated, k — equations of as. simple a form as possible to replace the & — i equations of the form yov+. . +y0n_xvn-x—y(y0u+ ■ ■ +2/°„_1w„-i) which would have existed had the roots been different. The most natural method^ of obtaining a suggestion lies probably in remarking that if r2 = rx + h, there is an integral [(x - £)ri+7,02 - (x - £)ri0i]/^> wherein the coefficients in 02 are the same functions of rxfih as are the coefficients in 1K, and (x - £)r10i by H, a circuit of the point £ changes K into K' = —^ [e27riri(x - £)ri0i + e2,riri (x - £)ri01(27ri + log (x - £))] = /qK + H. A similar artifice suggests itself when three of the roots of the determinantal equation are the same, and so on. We are thus led to the result, which is justified by an examination of the algebraic conditions, that whatever may be the circumstances as to the roots of the determinantal equation, n integrals exist, breaking up into batches, the values of the constituents H1( H2, • • • of a batch after circuit about x = £ being Hi' = /qHu + H3' = Ju1rH3 + H2, and so on. And this r is found to lead to the forms (x £) i0, (x £) i[0 1 1 + <px log (x - £)], (x - £)ri[xi + X2 log (»-£) + <Pi 0°g (z - £))2], and so on. Here each of 0i’/'iXiX • • is a series of positive and negative integral powers of x - £ in 2which the number of negative powers may be infinite. It appears natural enough now to inquire whether, under proper conditions for the forms of the rational functions ax ■ • an, it may be possible to ensure that in each of the series 0101Xi; • pezuiar the number of negative powers shall be finite. Herein equatioaSm lies, in fact, the limitation which experience has shown to be justified by the completeness of the results obtained. Assuming n integrals in which in each of ipx, fix, Xi * • the number of negative powers is finite, there is a definite homogeneous linear differential equation having nthese integrals found nby:! -1 nforming it to have n the form y' =(x-£') &1i/'( b + (x-£) &2y'< ~ > Conversely, + . . - + (x - ^)~ bny, where bx- • bn are finite for x=£. assume the equation tor have this form. Then2 on substituting a series of the form (x- £) [l + A^x- £) + A2(x- £) + • •] and equating the coefficients of like powers of x - £, it is found that r must be a root of an algebraic equation of order n ; this equation, which we shall call the index equation, can be obtained at once by substituting for y only (x-£)r and replacing each of bx • • bn by their values at x = £ ; arrange the roots rx, t2, • • of this equation so that the real part of r,- is equal to, or greater than, the real part of ri+1, and take r equal to rx; it is found that the coefficients A1; A„ • • are uniquely determinate, and that the series converges within a circle about x = £ which includes no other of the points at which the rational functions ax- • an become infinite. We have thus a solution H1 = (x —£)ri0i of the differential equation. If we now substitute in the equation y = Hxfyd,x, it is n_1 found to reduce to an equation of order 1 for 77 of the form 77' = (x-£)_1c1V(" 2) + . . . + (£ - fi)n~lcn_xr}, wherein cx ■ • cn_j are not infinite at x=£. To this equation precisely similar reasoning can then be applied ; its index equation has in fact the roots r2 — rx — l, • • ■, rn — rx—l; if r2 - rx be zero, the integral (x - £)“ Vi of the y equation will give an integral of the original equation containing log (x - £) ; if r2 - rx be an integer, and therefore a negative integer, the same will be true, unless in fix the term in (x - £)ri~r2 be absent; if neither of these arise, the original equation will have an integral (x - £),-’02. The y equation can now, by means of the one integral of it belonging to the index r2-rx-l, be similarly reduced to one of order ti-2, and so on. The result will be that stated above. We shall say that an equation of the form in question is regular about x = £. We may examine in this way the behaviour of the integrals at all the points at which any one of the rational functions ax • • an becomes infinite; in general we must expect that pucttsian beside these the value x= 00 will be a singular point for cquaplons_ the solutions1 of the differential equation. To test this we put x= /2 throughout, and examine as before at 2 = 0. For instance, the ordinary linear equation with constant coefficients has no singular point for finite values of x; at x= co it has a singular point2 and is not regular ; or again, Bessel’s equation x*y" + XT/’ + (x — 7i2)t/ = 0 is regular about x = 0, but not about x= . An equation regular at all the finite singularities and also at x= co is called a Fuchsian equation. We proceed to examine particularly the case of an equation of the second order y + ay + by—0. Putting x=^> ^ becomes d^y/dfi+ftt^ — at 2)dy/dt + bt~4y = Q, which is not regular about 2=0 unless 2 —at
 * this is -2

1

and bt 2,