See also:system of linear partial homogeneous differential equations of the first
See also:order, to the solution of which the various differential equations discussed have been reduced . It will be enough to consider whether the given differential equations allow the infinitesimal de— cIdxi— ... —(prdxr—pr+idxr+i ... —pn dxn into the
See also:form v(d f'—ee +id r+i — ... w°din) ; and here ass', ... cos may be taken, as before, to be
See also:principal integrals of a certain
See also:complete system of linear equations; those, namely, determining the characteristic chains. s If L be a
See also:function of t and of the 2fl quantities xi, . . x,, xl, . . xn, where x,denotes dx;/dt, &c., and if in the n equations d ((dL\l _ dL dt dx; 'dL we put p; = a x - , and so
See also:express 1 ,...; ' e ,n in terms of t, x;, . . . x,, I, . . . is., assuming that the
See also:determinant of the quantities &le dx,dr; is not zero; if, further, H denote the function of t, . . x,, . . . pn, numerically equal to Pixi+ . .+pnn—L, it is easy tions to prove that dp;/dt q—dH/dx;, dx,/dt=dI-I/dp; .
These Equa so-called canonical equations form
See also:part of those for of dynamics. the characteristic chains of the single partial equation dz/dt+H(t, xi, . xn, dz/dxi, . . dz/dx,,) =o, to which then the solution of the
See also:original equations for xi . . . x,, can be reduced . It may be shown (I) that if z= p(t, . . . xn, ci, Gn) +-c be a complete integral of this equation, then p;=dp/dx;, dp/dc:=e; are 2n equations giving the solution of the canonical equations referred to, where ci ... cn and el, ... en are arbitrary constants; (2) that if x = X; (t, x°i, . . Pe') pi = P; (t, xi, ... p°n) be the principal solutions of the canonical equations for t=t°, and w denote the result of substituting these values in pidH/dpi+ . . .+pndHtdp„=H, and S2= f te,wdt, where, after integration, S2 is to be expressed as a function of t, xi . . xn, . . xn°, then z = Sl+z° is a complete integral of the partial equation . A system of differential equations is said to allow a certain continuous
See also:group of transformations (see GRouPs, THEORY OF) when the introduction for the variables in the differen-Apptica tial equations of the new variables given by the t of theory of equations of the group leads, for all values of the cantina- parameters of the group, to the same differential equaousgrouPs tions in the new variables . It would be interesting to format to verify in examples that this i theories s the case in at least . the majority of the differential equations which are known to be integrable in finite terms .
We give a theorem of verygeneral application for the case of a simultaneous completetransformations of the group . It can be shown easily that sufficient conditions in order that a complete system Iilf =o ... Hsf =o, in n
See also:independent variables, should allow the infinitesimal transformation Pf =o are expressed by k equations II:Pf—PII;f=Xrlllif+...+X;xl sf . Suppose now a complete system of n—r equations in n variables to allow a group of r infinitesimal transformations (Pif, , P,f) which has an invariant subgroup of r—I parameters (Pif, P,_if), it being supposed that the n quantities IIif, . .., H,;_,f, Pif, . ., Pr are not connected by an identical linear equation (with coefficients even depending on the independent variables) . Then it can be shown that one solution of the complete system is deter-minable by a quadrature . For each of II,Pvf—PselI;f is-a linear function of lI,._rf and the simultaneous system of inde- pendent equations Ilif=o, II._rf=o, Pif=o, . . Pr_if=o is therefore a complete system, allowing the infinitesimal trans-formation Prf . This complete system of n—i equations has there-fore one
See also:common solution w, and Pr(w) is a function of w . By choosing w suitably, we can then make Pr(w)=I . From this equation and the n—I equations H te=o, Pas= o, we can determine w by a quadrature only .
Hence can be deduced a much more general result, that If the group of r parameters be integrable, the complete system can be entirely solved by quadratures; it is only necessary to introduce the solution found by the first quadrature as an independent variable, whereby we obtain a complete system of n—r equations in n-I variables, subject to an integrable group of r—I parameters, and to continue this
See also:process . We give some examples of the application of the theorem . (I) If an equation of the first order y' = p(x, y) allow the infinitesimal transformation ndf/dx-[-ndf/dy, the integral curves w(x, y) =y°, wherein w(x, y) is the solution of dx+p(x,y)dy=o reducing to y for x=x°, are interchanged among themselves by the infinitesimal transformation, or w(x, y) can be chosen to make Edw/dx+edw/dy = I ; this, with
See also:dw/dx+pdw/dy=o, determines w as the integral of the complete differential (dy—pdx)/(n—p ) . This result itself shows that every ordinary differential equation of the first order is subject to an infinite number of infinitesimal transformations . But every infinitesimal transformation fdf/dx+ndf/dy can by
See also:change of variables (after integration) be brought to the form df/dy, and all differential equations of the first order allowing this group can then be reduced to the form F(x, dy/dx) =o . (2) In an ordinary equation of the second order y" = 0(x, y, y'),
See also:tody/dx = yi, dyi/dx = p(x, y,yi), if H,Hi be the solutions for y and yi chosen to reduce to and when x =x°, and the equations H =y, Hi =yi be equivalent to w =y°, wi =ye, then co, wi are the principal solutions of
See also:IIf=dfldx+yidf/dy+pdfidyl =o . If the original equation allow an infinitesimal transformation whose first extended form (see GROUPS) is--Pf = fd f/dx+ndf/dy±nidf/dyi, where ni&t is the increment of dy/dx when t&t, not are the increments of- x, y, and is to be expressed in terms of x, y, yi, then each of Pw and
See also:Pen must be functions of w and to, or the partial differential equation IV must allow the group Pf . Thus by our general theorem, if the differential equation allow a group of two parameters (and such a group is always integrable), it can be solved by quadratures, our explanation sufficing, however, only provided the form IIf and the two infinitesimal transformations are not linearly connected . It can be shown, from the fact that ni is a quadratic polynomial in yi, that no differential equation of the second order can allow more than 8 really independent infinitesimal transformations, and that every homogeneous linear differential equation of the second order allows just 8, being in fact reducible to d2y/dx' =o . Since every group of more than two parameters has subgroups of two parameters, a differential equation of the second order allowing a group of more than two parameters can, as a
See also:rule, be solved by quadratures . By transforming the group we see that if a differential equation of the second order allows a single infinitesimal transformation, it can be transformed to the form F(x,dy/dx, d2y/dx2); this is not the case for every differential equation of the second order . (3) For an ordinary differential equation of the third order, allowing an integrable group of three parameters whose infinitesimal transformations are not Iinearly connected with the partial equation to which 'the solution of the given ordinary equation is reducible, the similar result follows that it can be integrated by quadratures .
But if the group of three parameters be
See also:simple, this result must be replaced by the statement that the integration is reducible to quadratures and that of a so-called Riccati equation of the first order, of the form dy/dx =A+By+Cy2, where A, B, C are functions of x . (q) Similarly for the integration by quadratures of an ordinary equation y,,= p(x, y, yi, . . . yn_i) of any order . Moreover, the group allowed by the equation may quite well consist of extended contact transformations . An important application is to the case where the differential equation is the resolvent equation defining the group , c¢ transformations or rationality group of another differential equation (see below) ; in particular, when the rationality group of an ordinary linear differential equation is integrable, the equation can be solved by quadratures . Following the
See also:practical and provisional division of theories of differential equations, to which we alluded at starting, into transformation theories and function theories, we pass now to give some account of the latter . These are both a necessary logical complement of the former, and the only remaining resource when the expedients of the former have been exhausted . While in the former investigations we have dealt only with values of the independent variables about which the functions are developable, the leading idea now becomes, as was long ago remarked by G .
See also:Green, the
See also:consideration of the neighbourhood of the values of the variables for which this developable character ceases . Beginning, as before, with existence theorems applicable for ordinary values of the variables, we are to consider the cases of failure of such theorems . When in a given set of differential equations the number of equations is greater than the number of dependent variables, the equations cannot be expected to have common solutions unless certain conditions of compatibility, obtainable by equating different forms of the same differential coefficients deducible from the equations, are satisfied . We have had examples in systems of linear equations, and in the case of a set of equations P1= ...
,Pr= 4, .. For the case when the number of equations is the same as that of dependent variables, the following is a general theorem which should be referred to: Let there be r equations in r dependent variables z', ... z, and n independent variables x', . . . xn; let the differential coefficient of za of highest order which enters be of order ha, and suppose dkaza/dx'ka to enter, so that the equations can be written d' cza/dxl''s = d'Q, where in the general differential coefficient of zp which enters in say dkl+ . . .. +knZp/dxlki . . . dxnkn, we have ki <hp and kid- . . . +kn hp . Let al, . . . a., b1, . . . b, and
See also:Oki . . . kn be a set of values of xl, . xn, Zi, .
. . Zr . and of the differential coefficients entering in 42a about which all the functions 401, . . . d'r are developable . Corresponding to each dependent variable za, we take now a set of ha functions of x2, ...xn, say Oa, ... , tba(h_r) arbitrarysave that they must be developable about
See also:a2, a3, . . . an, and such that for these values of x2, . . . the function Op reduces to bp, and the differential coefficient dk2+ . . . +kn4, ki)/dx2k2 . . . dxnkn reduces to bpki . . . kn .
Then the theorem is that there exists one, and only one, set of functions zi, . . . zr of x2, ... xn developable about al, . . . an satisfying the given differential equations, and such that for xi = al we have z = dz dx = OP), dha lz /dkQ ix _ 00.01a—1 ) And, moreover, if the arbitrary functions cba, cbQ i) ... contain a certain number of arbitrary variables tl, ... t,n, and be de- velopable about the values . . . t,n° of these variables, the solutions zl, ... Zr will contain . . . t,,,, and be developable about . . . t„,° . The
See also:proof of this theorem may be given by showing that if ordinarypower series in x1—al, . . xn—an, . t,.—t,, be substituted in the equations wherein in za the coefficients of (xi—a1)°, xl—a1, ., (xi—a1)hs 1 are the arbitrary functions dia, dia('), . . . dia(h-1), divided respectively by i, i!, 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
See also:analytical functions (see FUNCTION), 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
See also: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 of the independent variables with a different value of the function, belonging, as we say, to a different branch of the function; and there is even no reason for singular assuming that the number of branches is finite, or that Paints" different branches have the same singular points and solutions. 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, being deter-
See also:mined 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 reversion of an elliptic integral, in solving the equation (as) 2 = (x —al) (x — a2) (x — as) (x —a,) ; about an ordinary value the right side is developable; if we put x-a1=112, the right side becomes developable about t1=o; if we put x =13, the right side of the changed equation is developable about t=o; it is quite easy to show that the integral reducing to a definite value x° for a value s° is obtainable by a series in integral
See also:powers; this, however, must be supplemented by showing that for no value 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 the completeness and generality with which they can Linear be discussed . We have seen that' if in the equations different d y f d x =y', dyl/dx = y2, . . . , dy,,_2/dx = yn 1, ma, equa- dY 1/dx =a0y+aW ly1 F lions with .' +aly -1, rational co . where a1, a2, . . , an are now to be taken to be rational efficients . functions of x, the value x=x° be one for which no one of these rational functions is infinite, and y°, y°1, . be quite arbitrary finite values, then the equations are satisfied by y =Y°u +y°iui + . . . +Y°n_1u,—1, where u, u1, . are functions of x, independent of y°n_l, developable about x=x°; this value of y is such that for x=x-° the functions y, Yi . y,rl reduce respectively to y°, yi°, . y°n_'; it can be proved that the region of existence of these series extends within a circle centre x° and
See also:radius equal to the distance from x° of the nearest point at which one of a1, . a,, becomes infinite . Now consider a region enclosing x°, and only one of the places, say 2, at which one of a1, . a„ becomes infinite .
When x is made to describe a closed
See also:curve in this region, including this point 2 in its interior, it may well happen that the continuations of the functions u, u1, . . . , u,,.1 give, when we have returned to the point x, values v, vl, . . so that the integral under consideration becomes changed to y°v+y1°v1+ . . +Y°n_1vn_i . At x° let this branch and the corresponding values of yi, . . . y._' be 0°, ,7°l, . . then, as there is only one series satisfying the equation and reducing to (n°, n°1, . . . n°n_1) for x=x°, and the coefficients in the differential equation are single-valued functions, we must have n°u+m°u'+ . . +n°n_'un_1=y°v+y°'vi+ . . + y°n_1vn_l; as this holds for arbitrary values of y°, . y°n-i, upon which u, . . . un_1 and v, .
. . anal do not depend, it follows that each of v, . . . vn_t is a linear function of u, . . . u„_1 with
See also:constant coefficients, say vi =Ailu+ . . A;nun i . Then y°v+ ... +Y°n_iVn-1= (E.AS1yi°) u+ ... +(F'iAiny °i)un_l; this is equal to µ(y°u+ . . +y°n-lun-1) if 2i.Airy° . =A31°r-1; eliminating y°, . . . y°n_i from these linear equations, we have a determinantal equation of order n forµ; letul be one of its roots; determining the ratios of y°, yi°, . y°,r1 to satisfy the linear equations, we have thus proved that there exists an integral, H, of the equation, which when continued
See also:round the point 2 and back to the starting-point, becomes changed to H1=µ1H . Let now l; be the value of x at and ri one of the values of (1/2lri)
See also:log µl; consider the function (x—E)—t'H; when x makes a circuit round x=E, this becomes changed to exp(—2xir1) (x—E)-rlisH, that is, is unchanged; thus we may put H = (x—E)rid,1, d11 being a function single-valued for paths in the region considered described about 2, and therefore, by
See also:Laurent's Theorem (see FUNCTION), capable of expression in the
See also:annular region about this point by a series of
See also:positive and negative integral powers of x—i;, which in general may contain an infinite number of negative powers; there is, however, no reason to suppose r1 to be an integer, or even real . Thus, if all the roots of the determinantal equation in ,a are different, we obtain n integrals of the forms (x—E)rig11, In general we obtain as many integrals of this form as there are really different roots; and the problem arises to discover, in case a
See also:root be k times repeated, k—i equations of as simple a form as possible to replace the k—i equations of the form y°v+ .
+ y°n_1vn_'=µ(y°u+ . . . y°n_lun_1) which would have existed had the roots been different . The most natural method of obtaining a
See also:suggestion lies probably in remarking that if
See also:r2=rl+h, there is an integral [(x—E)ri+kt2-(x-E)riO1]/h, where the coefficients in 41s are Consideration of function theories of differential equations . A general existence theorem . the same functions of ri+h as are the coefficients in 01 of r1; when h vanishes, this integral takes the form (x—t)'1[d l+1611og (x-0] , or say (x—5)'1[01+W1 log (x—E)]; denoting this by 2xiislK, and (x—¢)rl 01 by H, a circuit of the point K'= I• [e2,ra=(x—E)'101+r(x—)*101(2iri+log (x—E))]=µ1K+H . 21rtsn 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 Hl, H2, . . . of a batch after circuit about x=E being Hl'=p1H1, H2'=,1H2+H1, Hs'=iziHi+H2, and so on . And this is found to lead to the forms (x-E)ri01, (x—E)'I[C +01 log (x —E)], (x—E)"'I[Xl+x2 log (x—E)+01 (log(x—E))21, and so on . Here each of 01, xi, x2, . . . is a series of positive and negative integral powers of x—E in which 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 al, .
. . a", it may be possible to ensure that in each of the series o1, 01, x1, .. .
See also:Regular the number of negative powers shall be finite . Herein equations. lies, in fact, the
See also:limitation which experience has shown to be justified by the completeness of the results obtained . Assuming n integrals in which in each of 0,, 01, xi ... the number of negative powers is finite, there is a definite homogeneous linear differential equation having these integrals; this is found by forming it to have the form t Y'"=(x—) +(x—t)-2b2y'("-2)+...+(x—'"b"y, where b1, . . -. b" are finite for x=E . Conversely, assume the equation to have this form . Then on substituting a series of the form (x—E)'[1+AI(x—E)+A2(x—tS)2+ .] and equating the coefficients of like powers of x —E, it is found that r must be a root of an algebraic equation of order n; this equation, which we shall
See also:call the
See also:index equation, can be obtained at once by substituting for y only (x—l;)' and replacing each of bl, b" by their values at x=;; arrange the roots r1, r2, ... of this equation so that the real part of r; is equal to, or greater than, the real part of r, . , and take r equal to Ti; it is found that the coefficients Al, A2 .. . are uniquely determinate, and that the series converges within a circle about xes i; which includes no other of the points at which the rational functions al, . . . as become infinite . We have thus a solution HI = (x — E)'i,¢i of the differential equation . If we now substitute in the equation y=Hlfndx, it is found to reduce to an equation of order n—i for n of the form n'('-1) = (.x–E) Icifl'(" + ...
+(x–E)"-Ic"-in, where c1, ... c„-1 are not infinite at x=E . To this equation precisely similar reasoning can then be applied; its index equation has in fact the roots r2—rl—i r"—rl—I; if r2—rl be zero, the integral (x— )-101 of the n equation will give an integral of the original equation containing log (x—i;); if r2—rl be an integer, and therefore a negative integer; the same will be true, unless in Wi the
See also:term in (x—E)'i-'2 be absent; if neither of these arise, the original equation will have an integral (x —E)r202 . Then equation can now, by means of the one integral of it belonging to the index r2—r1–i, be similarly reduced to one of order n -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 al . . . as Fuchsian becomes infinite; in general we must expect that beside equations. these the value x=oo will be a singular point for the solutions of the differential equation . To test this we put x I/t throughout, and examine as before at t=o . 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 point and is not regular; or again, Bessel's equation x2y"+xy'+(x2—n2)y=o is regular about x = o, but not about x = w . An equation regular at all the finite singularities and also at x--= oo is called a Fuchsian equation . We proceed to examine particularly the case of an equation of the second order Y"+ay'+bY=o . Putting x = lit, it becomes d2y/dt2+(2r' —at-2)dy/dt + bt-'y = o, which is not regular about t=o unless 2 —at-1 and bt2, that is, unless ax and bx2 are finite at x = oo ; which we thus assume; putting Bquatlon y=tr(i+Alt+ .. .), we find for the index equation atx=o0 of the _ the equation r(r—I)+r(2—ax)o+(bx2)o=o .
If there be second finite singular points at ¢i, . F,", where we assume order. m> t, the cases m = o, m= I being easily dealt with, and if 0(x)=(x—El) . . . (x—i;,,,), we must have a.:1)(x) and b.[0(x)]2 finite for all finite values of x, equal say to the re- spective polynomials 0(x) and 8(x), of which by the conditions at x= oo the highest respective orders possible are m -1 and 2(M –I) . The index equation atx= El is r(r I)+0(6)/4'(Ei)+0(0I/[O'(El)]2 =o, and if al, S1 be its roots, we have al+si = I –O(Ei)/e'(fI) and ai$i=0(E1)/[4'(E1)12• Thus by an elementary theorem of
See also:algebra, the sum E(I–a,–S,)/(x—f,), extended to the m finite singular points, is equal to 0(x)/, (x), and the sum E(I —ai—Si) is equal to the ratio of the coefficients of the highest powers of x in 0(x) and 0(x), and therefore equal to I+a+f , where a, S are the indices at x= so . Further, if (x, I),,,_2 denote the integral part of the quotient B(x)/c)(x), we have Ea, 4'(,)I(x— equal to — (x, I),,,-2+e(x)/sb(x), and the coefficient of x'"-2 in (x, I),,,_2 is as . Thus the differential equation has the form Y"+y'I (I —a, -13,)/(x-6=) +Y[(x, I)m_2/(x—W I/0(x) = o . If, however, we make a change in the dependent variable, putting y=(x—6i)"i . . . (x—E,,,)"mn, it is easy to see that the equation changes into one having the same singular points about each of which it is regular, and that the indices at x = l;, become o and S, — an which we shall denote by X,, for (x—i;/)'j can be
See also:developed in positive integral powers of x—E, about x=i;,; by this transformation the indices at .x= oo are changed to 1 a+al+ ..
See also:Tam, YqR+Npl+..+Sm which we shall denote by A, p . If we suppose this change to have been introduced, and still denote the independent variable by y, the equation has the form y"+y'z(I —X,)/(x—s,)+y(x, I),,,_2/0(4 =o, while X+p+Xi+ ... +Xm=m–I .
Conversely, it is easy to verify that if Xis be the coefficient of x'"-2 in (x, i)m_2, this equation has the specified singular points and indices whatever be the other coefficients in (x, I)m_2 . Thus we see that (beside the cases m =o, m= I) the " Fuchsian equation" of the second order with two finite singular points is distinguished by the fact that it has a definite form when the singular points and the indices are assigned. lfypergeo-In that case, putting (x—Ei)/(x—62) =1/(1—I), the singular metric points are transformed to o, , co, and, as is clear, without equation. change of indices . Still denoting the independent variable by x, the equation then has the form x(I–x)y"+y'[I—AI—x(I+X+p)1—XpY=o, which is the ordinary hypergeometric equation . Provided none of XI, X2, X—p be zero or integral about x=o, it has the solutions F(X, p, I —Xi, x), xA1F(X+Ai, p+Xl, I+Al, x) ; about x=i it has the solutions F(X, p, I —A2, I –x), (I —x)X2F(X+X2, p+X2, +X2, I –x), where X+p+Xi+X2 =I; about x = oo it has the solutions x AF(X, X+Xi, X–p+I, x-I), xT 'F(p, p+Xl, x-'), where F(a, S, y, x) is the series I+asx+a(a+I)S(S+i)x2 y I.2.7(7+1) which converges when ix ! < i, whatever a, S, y may be, converges for all values of x for which Ix ! = I provided the real part of y — a — S < o algebraically, and converges for all these values except x = I provided the real part of y—a-a> —i algebraically . In accordance with our general theory, logarithms are to be expected in the solution when one of XI, X2, X—p is zero or integral . Indeed when Al is a negative integer, not zero, the second solution about x =o would contain vanishing factors in the denominators of its coefficients; in case A or p be one of the positive integers I, 2, . . . (—XI), vanishing factors occur also in the numerators; and then, in fact, the. second solution about x=o becomes xAi times an integral polynomial of degree (—XI) —X or of degree (—XI)—p . But when XI is a negative integer including zero, and neither A nor is is one of the positive integers I, 2 . . (-AI), the second solution about x =o involves a term having thefactor log x .
When XI is a positive integer, not zero, the second solution about x=o persists as a solution, in accordance with the order of arrangement of the roots of the index equation in our theory; the first solution is then replaced by an integral polynomial of degree—A or—p, when A or ).i is one of the negative integers o,—I,—2, . . ., I —XI, but otherwise contains alogarithm . Similarly for the solutions about x=i or x=oo ; it will be seen below how the results are deducible from those for x = o . Denote now the solutions about x=o by u1, u2; those about x= I by vl, v2; and those about x=oo by wi,
See also:w2; in the region (S Si) common to the circles S", SI of radius i whose centres are the points x=o, x=I, all the first four are valid,
See also:march and there exist equations u1=Avi+Bv2, u2=Cvi+Dv2 ofthe where A, B, C, D are constants; in the region (
See also:S1S) integral lying inside the circle SI and outside the circle So, those that are valid are v1, V2, WI, w2, and there exist equations sr = PwI+Qw2, v2=Rwi+Tw2, where P, Q, R, T are constants; thus considering any integral whose, expression within the circle So is aul+bu2, where a, b are constants, the same integral will be represented within the circle SI by (aA+bC)vi+(¢B+bD)v2, and outside these circles will be represented by [(aA+bC) P+(aB +bD)R]wI+[(aA+bC)Q+(aB+bD)T]w2 . A single-valued branch of such integral can be obtained by making a barrier in the
See also:plane joining co to o and I to m ; for instance, by excluding the consideration of real negative values of x and of real E changes K into positive values greater than I, and defining the phase of x and x-I for real values between o and I as respectively o and a . We can form the Fuchsian equation of the second order with three arbitrary singular points I, z 2, Es, and no singular point at x = co , wed with respective indices a1, SI, a2, ~2, as, #3 such that al+i+a2+Q2+as+Qs = I . This equation can then be transformed into the hypergeometric equation in 24 ways; for out of El, E2, s we can in six ways choose two, say Ei, E2, which are to be transformed respectively into o and I, by (x-5I)/(x-Es) =t(1-I); and then there are four possible transformations of the dependent variable which will reduce one of the indices at t=o to zero and one of the indices at t = I also to zero, namely, we may reduce either al or /~ ,+I at t=o, and simultaneously either as or #2 at t=l . Thus the hypergeometric equation itself can be transformed into itself in 24 ways, and from the expression F(X, u, I-h1, x) which satisfies it follow 23 other forms of solution; they involve four series in each of the arguments, x, I/x, I/(I-x), (x-I)/x, x/(x-I) . Five of the 23 solutions agree with the fundamental solutions already described about x = o, x = I , x = co ; and from the principles by which these were obtained it is immediately clear that the 24 forms are, in value, equal in fours . The quarter periods K, K' of
See also:Jacobi's theory of elliptic functions, of which K= fo /2(1-h sin 29) dd, and K' is the same function of I-h, can easily be proved to be the solutions of a hypergeometric equation of which h is the independent variable . When K, K' are regarded as defined in terms of h by the differential modular a equation, the ratio K'/K is an infinitely many valued functions. function of h . But it is remarkable that Jacobi's own theory of theta functions leads to an expression for h in terms of K'/K (see FUNCTION) in terms of single-valued functions .
We may thenattempt to investigate, in general, in what cases the independent variable x of a hypergeometric equation is a single-valued function of the ratios of two independent integrals of the equation . The same inquiry is suggested by the problem of ascertaining in what cases the hypergeometric series F(a, 0, y, x) is the expansion of an algebraic (irrational) function of x . In order to explain the meaning of the question, suppose that the plane of x is divided along the real
See also:axis from -co to o and from I to +co, and, supposing logarithms not to enter about x =o, choose two quite definite integrals yI, y2 of the equation, say yi = F(a, tr, I-XI, x), y2 =x1iF(X+Xi, it+al, I +XI, x), with the
See also:condition that the phase of x is zero when x is real and between o and I . Then the value of s=y2/yi is definite for all values of x in the divided plane, s being a single-valued monogenic branch of an analytical function existing and without singularities all over this region . If, now, the values of s that so arise be plotted on to another plane, a value p+iq of s being represented by a point (p, q) of this s-plane, and the value of x from which it arose being mentally associated with this point of the s-plane, these points will fill a connected region therein, with a continuous boundary formed of four portions corresponding to the two sides of the two barriers of the x-plane . The question is then, firstly, whether the same value of s can arise for two different values of x, that is, whether the same point (p, q) of the s-plane can arise twice, or in other words, whether the region of the s-plane overlaps itself or not . Supposing this is not so, a second part of the question presents itself . If in the x-plane the barrier joining - oo to o be momentarily removed, and x describe a small circle with centre at x=o starting from a point x= -h-ik, where h, k are small, real, and positive and coming back to this point, the original value s at this point will be changed to a value o, which in the original case did not arise for this value of x, and possibly not at all . If, now, after restoring the barrier the values arising by continuation from a be similarly plotted on the s-plane, we shall again obtain a region which, while not overlapping itself, may quite possibly overlap the former region . In that case two values of x would arise for the same value or values of the quotient y2/yi, arising from two different branches of this quotient . We shall understand then, by the condition that x is to be a single-valued function of x, that the region in the s-plane corresponding to any branch is not to overlap itself, and that no two of the regions corresponding to the different branches are to overlap . Now in describing the circle about x=o from x = -h-ik to -h+ik, where h is small and k evanescent, s=xajF(a+XI, u+a1, +X1, x)f (%, µ, I-Xi, x) is changed to a=see,riAl .
Thus the two portions of boundary of the s-region corresponding to the two sides of the barrier (-so, o) meet (at s =o if the real part of XI be positive) at an
See also:angle 2srLi, where LI is the absolute value of the real part of XI; the same is true for the a-region representing the branch a . The condition that the s-region shall not overlap itself requires, then, LI=1 . But, further, we may form an infinite number of branches o=se2''iAl, ol=eeniki, . . in the same way, and the corresponding regions in the plane upon which y2/yi is represented will have a common point and each have an angle 2srL1; if neither overlaps the preceding, it will happen, if LI is not zero, that at length one is reached overlapping the first, unless for some positive integer a we have 2,raLI=2,r, in other words L1=I/a . If this be so, the branch o4-i=se2ri4AI will be represented by a region having the angle at the common point common with the region for the branch s; ,but not altogether coinciding with this last region unless XI be real, and therefore ==I/a; then there is only a finite number, a, of branches obtainable in this way by
See also:crossing the barrier (—co , o) . In precisely the same way, if we had begun by taking the quotient s'= (x—I)12F(a+X2,,i+X2, I+X2;1-x) IF(X, u, I-)i2, I—x) of the two solutions about x =I, we should have found that x is not a single-valued function of s' unless a2 is the inverse of an integer, or is zero; as s' is of the form (As+B)I (Cs+D), A, B, C, D constants, the same is true in our case; equally, by considering the integrals about x=so we find, as a third condition necessary in order that x may be a single-valued function of s, that X- u must be the inverse of an integer or be zero . These three differences of the indices, namely, XI, A2, X- s, are the quantities which enter in the differential equation satisfied by x as a function of s, which is easily found to be xni 3x2n = I +~(h-hl-h2)x-i (x—I)—I +ahix-2 +02 (x—I)-2, x13 ?x1 where xi =dx/ds, &c.; and hi = I-y12, h2 =1-X24, ha= i-(X—µ)e . Into the converse question whether the three conditions are sufficient to ensure (I) that the s region corresponding to any branch does not overlap itself, (2) that no two such regions overlap, we have no space to enter . The second question clearly requires the inquiry whether the group (that is, the monodromy group) of the differential equation is properly discontinuous . ( See GROUPS, THEORY OF.) The foregoing account will give an idea of the nature of the function theories of differential equations; it appears essential not to exclude some explanation of a theory intimately related both to such theories and to transformation theories, which is a generalization of
See also:Galois's theory of algebraic equations . We
See also:deal only with the application to homogeneous linear differential equations . In general a function of variables xi, x2 . is said to be rational when it can be formed from them and the integers I, 2, 3, . by a finite number of additions, subtractions, multiplications Rationality and divisions .
We generalize thisdefinition . Assume that group of we have assigned a fundamental series of quantities and a linear functions of x, in which x itself is included, such that all equation. quantities formed by a finite number of additions, subtrac- tions, multiplications, divisions and differentiations in regard to x, of the terms of this series, are themselves members of this series . Then the quantities of this series, and only these, are called rational . By a rational function of quantities y, q, r, .. is meant a function formed from them and any of the fundamental rational quantities by a finite number of the five fundamental operations . Thus it is a function which would be called, simply, rational if the fundamental series were widened by the addition to it of the quantities p, q, r, . and those derivable from them by the five fundamental operations . A rational ordinary differential equation, with x as independent and y as dependent variable, is then one which equates to zero a rational function of y, the order k of the differential equation being that of the highest differential coefficient yk' which enters; only such equations are here discussed . Such an equation P = o is called irreducible when, firstly, being arranged as an integral polynomial in y(k), this polynomial is not the product of other polynomials in yk) also lrreduci- of rational form; and, secondly, the equation has no hilityof a solution satisfying also a rational equation of
See also:lower order. rational From this it follows that if an irreducible equation Po equation. have one solution satisfying another rational equation Q = o of the same or higher order, then all the solutions of P = o also satisfy Q.= o . For from the equation P = o we can by differentiation express y(k+i) y(k+2), . in terms of x, y, y'), . . . , yk), and so put the function Q rationally in terms of these quantities only . It is sufficient, then, to prove the result when the equation Q = o is of the same order as P =o .
Let both the equations be arranged as integral polynomials in y(k); their algebraic eliminant in regard to yk? must then vanish identically, for they are known to have one common solution not satisfying an equation of lower order; thus the equation P =o involves Q =o for all solutions of P = o . Now let y")=aiy"-1)+ . . . +a„y be a given rational homogeneous linear differential equation; let yi, . . . y,, be n particular functions of x, unconnected by any equation with constant coefficients of the form c1y1+ . . . +c„y„ =o, all satisfying The the differential equation; let . . . q„ be linear functions variant of yI, . . . ya, say ,7; =A;,y1+ . • • +AieYe, where the function constant coefficients Ai; have a non-vanishing deter- for a minant; write (q) =A(y), these being the equations of a linear general linear homogeneous group whose transformations equation . may be denoted by A, B, . . We
See also:desire to form a rational function't(n), or say t(A(y)), of ,, . n, in which the re constants Ai; shall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the yI, ... y, were connected by a linear equation with constant coefficients .
Such a function is in fact given, if the solutions Yi, . . . y, be developable Transformation of the equation into itself . in positive integral powers about x=a,by0(n)=nl+(x-a)"ni+ ... + (x–a)t"–1i"n" . Such a function, V, we call a variant . Then differentiating V in regard to x, and replacing ni(n) by its value a,n('–1) -F . . +a"n, we can arrange dV/dx, and similarly each of d'V/dx' . dNV/dxN, where N=n', as a linear function of The re- the N quantities m, • n", . . ni("--1) . . . n"<"–1i; and lvent thence by elimination obtain a linear differential equation so solve t for V of order N with rational coefficients . This we denote by F =o . Further, each of ni, . . .n" is expressible as a linear function of V, dV/dx, .
. dN–1V/dxN-1, with rational coefficients not involving any of the n1 coefficients Ail, since otherwise V would satisfy a linear equation of order less than N, which is impossible, as it involves (linearly) the n' arbitrary coefficients which would not enter into the coefficients of the supposed equation . In particular, yi, . y,, are expressible rationally as linear functions of w, dwldx, . dN–Iw/dxN–', where co is the particular function 4.(y) . Any solution W of the equation F=o is derivable from functions i'1, . . 1'", which are linear functions of yi, . y", just as V was derived from nl, . . .n,,; but it does not follow that these functions . . .snare obtained from yi, . . . y" by a transformation of the linear group A, B, . . . ; for it may happen that. the determinant d(ri, . . . g,,)/(dyi, . y,,) is zero . In that case D 1; . may be called a singular set, and W a singular solution; it satisfies an equation of lower than the N-th order . But every solution V, W, ordinary or singular, of the equation F=o, is expressible rationally in terms of w, dw/dx, .
. dN-1w/dxN–1; we shall write, simply, V =r(w) . Consider now the rational irreducible equation of lowest order, not necessarily a linear equation, which is satisfied by w; as yi, y" are particular functions, it may quite well be of order less than N ; we call it the resolvent equation, suppose it of order p, and denote it by 7(v) . Upon it the whole theory turns . In the first place, as 7(v) =o is satisfied by the solution w of F =o, all the solutions of 7(v) are solutions F =o, and are therefore rationally expressible by w; any one may then be denoted by r(w) . If this solution of F=o be not singular, it corresponds to a transformation A of the linear group (A, B, . .), effected upon yi, . y,, . The coefficients Ai; of this transformation follow from the expressions before mentioned for ni . ..n,, in terms of V,dV/dx,d'V/dx', . . . by substituting V=r(w); thus they depend on the p arbitrary parameters which enter into the general expression for the integral of the equation y(v) =o . Without going into further details, it is then clear enough that the resolvent equation, being irreducible and such that any solution is expressible rationally, with p parameters, in terms of the solution w, enables us to define a linear homogeneous group of transformations of yl . . . y,, depending on p parameters; and every operation of this (continuous) group corresponds to a rational transformation of the solution of the resolvent equation . This is the group called the rationality group, or the group of trans-formations of the original homogeneous linear differential equation .
The group must not be confounded with a subgroup of itself, the monodromy group of the equation, often called simply the group of the equation, which is a set of transformations, not depending on arbitrary variable parameters, arising for one particular fundamental set of solutions of the linear equation see GROUrs, THEORY OF) . The importance of the rationality group consists in three propositions . (I) Any rational function of y', . . 'which is unaltered in The fun- value by the transformations of the group can be written The fu in rational form . (2) If any rational function be changed
See also:dame t in form, becoming a rational function of yi, . . . y,,, a theorem 'n retard transformation of the group applied to its new form will to the leave its value unaltered . (3) Any homogeneous linear ration- transformation leaving unaltered the value of every nifty rational function of yi, " which has a rational value, group . belongs to the group . It follows from these that any group of linear homogeneous transformations having the properties 1) (2) is identical with the group in question . It is clear that with these properties the group must be of the greatest importance in attempting to discover what functions of x must be regarded as rational in order that the values of yi . . . y" may be expressed . And this is the problem of solving the equation from another point of view .
S .Lie und G . Scheffers, Geometrie der Berfihrungstransformationen, Bd. i . (
See also:Leipzig, 1896) ; Forsyth, Theory of Differential Equations, Part i., Exact Equations and
See also:Pfaff's Problem (Cambridge, 189o); S . Lie, "Allgemeine Untersuchungen caber Differentialgleichungen, die eine continuirliche endliche Gruppe gestatten " (Memoir), Mathem . Annal.
See also:xxv . (1885), pp . 71-151; S . Lie und G . Scheffers, Vorlesungen fiber Differentialgleichungen mit bekannten infinitesimalen Transformationen (Leipzig, 1891) . A very full bibliography is given in the
See also:book of E. v . Weber referred to; those here named are perhaps sufficiently representative of
See also:works .
Of classical works may be named: Jacobi, Vorlesungen fiber Dynamik (von A . Clebsch,Berlin, 1866) ; Werke, Supplementband; G
See also:Monge, Application de l'analyse d la geometrie (
See also:par M . Lionville,
See also:Paris, I85o); J . L .
See also:Lagrange, Legonssur le calcul
See also:des fonctions (Paris, 18o6), and Theorie des fonctions analytiques (Paris, Prairial, an V) ; G .
See also:Boole, A
See also:Treatise on Differential Equations (
See also:London, 1859); and Supplementary
See also:Volume (London, 1865); Darboux, Legons sur la theorie generale des surfaces, tt. i.-iv . (Paris, 1887–1896) ; S . Lie, Theorie der transformationsgruppen ii . (on Contact Transformations) (Leipzig, 1890) . (B) Quantitative or Function Theories for Linear Equations:—C .
See also:Jordan, Cours d'analyse, t. iii . (Paris, 1896) ; E .
Picard, Traite d'analyse, tt. ii. and iii . (Paris, 1893, 1896) ; Fuchs, Various
See also:Memoirs, beginning with that in Crelle's Journal, Bd. lxvi. p . 121; Riemann, Werke, 2r Aufi . (1892) ; Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen, Bde. i.-ii . (Leipzig, 1895–1898); Heffter, Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhdngigen Variablen (Leipzig, 1894);
See also:Klein, Vorlesungen caber lineare Diflerentialgleichungen der zweiten Ordnung (Autographed,
See also:Gottingen, 1894) ; and Vorlesungen fiber die hypergeometrische Function (Autographed, Gottingen, 1894); Forsyth, Theory of Differential Equations, Linear Equations . (y) Rationality Group (of Linear Differential Equations) :—Picard, Traite d'Analyse, as above, t. iii.; Vessiot, Annales de l'Ecole Normale, serie III. t. ix. p . 199 (Memoir) ; S . Lie, Transformationsgruppen, as above, iii . A connected account is given in Schlesinger, as above, Bd. ii., erstes Theil . (S) Function Theories of Non-Linear Ordinary Equations:—Painleve, Legons sur la theorie analytique des equations differentielles (Paris, 1897, Autographed) ; Forsyth, Theory of Differential Equations, Part ii., Ordinary Equations not Linear (two volumes, ii. and iii.) (Cambridge, 1900) ; KOnigsberger, Lehrbuch der Theorie der Differentialgleichungen (Leipzig, 1889); Painleve, Legons sur l'integration des equations differentielles de la mecanique et applications (Paris, 1895) . (e) Formal Theories of Partial Equations of the Second and Higher Orders:—E . Goursat, Legons sur l'integration des equations aux derivees partielles du second ordre, tt. i. and ii .
(Paris, 1896, 1898); Forsyth, Treatise on Differential Equations (London, 1889) ; and Phil . Trans .
See also:Roy .
See also:Soc . (A.), vol. cxci . (1898), pp . 1-86, (g) See also the six extensive articles in the second volume of the German
See also:Encyclopaedia of
See also:Mathematics . (H . F .
CALCULUS OF DIFFERENCES (Theory of Finite Differenc...
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