dz/dt − pdx/dt − qdy/dt = 0,
as we know, but also U = dz/du − pdx/du − qdy/du is also zero. For we have
which is equal to
As dF/dz is a developable function of t, this, giving
shows that U is everywhere zero. Thus integrals of F = 0 are obtainable by considering the aggregate of characteristic chains issuing from arbitrary chain connectivities T satisfying F = 0; and such connectivities T are, it is seen at once, determinable without integration. Conversely, as such a chain connectivity T can be taken out from the elements of any given integral all possible integrals are obtainable in this way. For instance, an arbitrary curve in space, given by x0= θ(u), y0= φ(u), z0= ψ(u), determines by the two equations F(x0, y0, z0, p0, q0) = 0, ψ′(u) = p0θ′(u) + q0φ′(u), such a chain connectivity T, through which there passes a perfectly definite integral of the equation F = 0. By taking ∞² initial chain connectivities T, as for instance by taking the curves x0= θ, y0= φ, z0= ψ to be the ∞² curves upon an arbitrary surface, we thus obtain ∞² integrals, and so ∞4elements satisfying F = 0. In general, if functions G, H, independent of F, be obtained, such that the equations F = 0, G = b, H = c represent an integral for all values of the constants b, c, these equations are said to constitute acomplete integral. Then ∞4elements satisfying F = 0 are known, and in fact every other form of integral can be obtained without further integrations.
In the foregoing discussion of the differential equations of a characteristic chain, the denominators dF/dp, ... may be supposed to be modified in form by means of F = 0 in any way conducive to a simple integration. In the immediately following explanation of ideas, however, we consider indifferently all equations F = constant; when a function of x, y, z, p, q is said to be zero, it is meant that this is so identically, not in virtue of F = 0; in other words, we consider the integration of F = a, where a is an arbitrary constant. In the theory of linear partial equations we have seen that the integrationOperations necessary for integration of F = a.of the equations of the characteristic chains, from which, as has just been seen, that of the equation F = a follows at once, would be involved in completely integrating the single linear homogeneous partial differential equation of the first order [Fƒ] = 0 where the notation is that explained above underContact Transformations. One obvious integral is ƒ = F. Putting F = a, where a is arbitrary, and eliminating one of the independent variables, we can reduce this equation [Fƒ] = 0 to one in four variables; and so on. Calling, then, the determination of a single integral of a single homogeneous partial differential equation of the first order in n independent variables,an operation of ordern − 1, the characteristic chains, and therefore the most general integral of F = a, can be obtained by successive operations of orders 3, 2, 1. If, however, an integral of F = a be represented by F = a, G = b, H = c, where b and c are arbitrary constants, the expression of the fact that a characteristic chain of F = a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] = 0, these three relations being identically true. Conversely, suppose that an integral G, independent of F, has been obtained of the equation [Fƒ] = 0, which is an operation of order three. Then it follows from the identity [ƒ[φψ]] + [φ[ψƒ]] + [ψ[ƒφ]] = dƒ/dz [ψφ] + dφ/dz [ψƒ] + dψ/dz [ƒφ] before remarked, by putting φ = F, ψ = G, and then [Fƒ] = A(ƒ), [Gƒ] = B(ƒ), that AB(ƒ) − BA(ƒ) = dF/dz B(ƒ) − dG/dz A(ƒ), so that the two linear equations [Fƒ] = 0, [Gƒ] = 0 form a complete system; as two integrals F, G are known, they have a common integral H, independent of F, G, determinable by an operation of order one only. The three functions F, G, H thus identically satisfy the relations [FG] = [GH] = [FH] = 0. The ∞² elements satisfying F = a, G = b, H = c, wherein a, b, c are assigned constants, can then be seen to constitute an integral of F = a. For the conditions that a characteristic chain of G = b issuing from an element satisfying F = a, G = b, H = c should consist only of elements satisfying these three equations are simply [FG] = 0, [GH] = 0. Thus, starting from an arbitrary element of (F = a, G = b, H = c), we can single out a connectivity of elements of (F = a, G = b, H = c) forming a characteristic chain of G = b; then the aggregate of the characteristic chains of F = a issuing from the elements of this characteristic chain of G = b will be a connectivity consisting only of elements of
(F = a, G = b, H = c),
and will therefore constitute an integral of F = a; further, it will include all elements of (F = a, G = b, H = c). This result follows also from a theorem given underContact Transformations, which shows, moreover, that though the characteristic chains of F = a are not determined by the three equations F = a, G = b, H = c, no further integration is now necessary to find them. By this theorem, since identically [FG] = [GH] = [FH] = 0, we can find, by the solution of linear algebraic equations only, a non-vanishing function σ and two functions A, C, such that
dG − AdF − CdH = σ(dz − pdz − qdy);
thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx + qdy and constitute a connectivity, which is therefore an integral of F = a. While, further, from the associated theorems, F, G, H, A, C are independent functions and [FC] = 0. Thus C may be taken to be the remaining integral independent of G, H, of the equation [Fƒ] = 0, whereby the characteristic chains are entirely determined.
When we consider the particular equation F = 0, neglecting the case when neither p nor q enters, and supposing p to enter, we may express p from F = 0 in terms of x, y, z, q, and then eliminate it from all other equations. Then instead of the equation [Fƒ] = 0, we have, if F = 0 give p = ψ(x, y, z, q), the equation
moreover obtainable by omitting the term in dƒ/dp in [p − ψ, ƒ] = 0. Let x0, y0, z0, q0, be values about which the coefficients inThe single equation F = 0 and Pfaffian formulations.this equation are developable, and let ζ, η, ω be the principal solutions reducing respectively to z, y and q when x = x0. Then the equations p = ψ, ζ = z0, η = y0, ω = q0represent a characteristic chain issuing from the element x0, y0, z0, ψ0, q0; we have seen that the aggregate of such chains issuing from the elements of an arbitrary chain satisfying
dz0= p0dx0− q0dy0= 0
constitute an integral of the equation p = ψ. Let this arbitrarychain be taken so that x0is constant; then the condition for initial values is only
dz0− q0dy0= 0,
and the elements of the integral constituted by the characteristic chains issuing therefrom satisfy
dζ − ωdη = 0.
Hence this equation involves dz − ψdx − qdy = 0, or we have
dz − ψdx − qdy = σ(dζ − ωdη),
where σ is not zero. Conversely, the integration of p = ψ is, essentially, the problem of writing the expression dz − ψdx − qdy in the form σ(dζ − ωdη), as must be possible (from what was said underPfaffian Expressions).
To integrate a system of simultaneous equations of the first order X1= a1, ... Xr= arin n independent variables x1, ... xnand one dependent variable z, we write p1for dz/dx1, &c.,System of equations of the first order.and attempt to find n + 1 − r further functions Z, Xr+1... Xn, such that the equations Z = a, Xi= ai,(i = 1, ... n) involve dz − p1dx1− ... − pndxn= 0. By an argument already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation Xi= ai; thus each of the expressions [XiXj] must vanish in virtue of the equations expressing the integral, and we may without loss of generality assume that each of the corresponding ½r(r − 1) expressions formed from the r given differential equations vanishes in virtue of these equations. The determination of the remaining n + 1 − r functions may, as before, be made to depend on characteristic chains, which in this case, however, are manifolds of r dimensions obtained by integrating the equations [X1ƒ] = 0, ... [Xrƒ] = 0; or having obtained one integral of this system other than X1, ... Xr, say Xr+1, we may consider the system [X1ƒ] = 0, ... [Xr+1ƒ] = 0, for which, again, we have a choice; and at any stage we may use Mayer’s method and reduce the simultaneous linear equations to one equation involving parameters; while if at any stage of the process we find some but not all of the integrals of the simultaneous system, they can be used to simplify the remaining work; this can only be clearly explained in connexion with the theory of so-called function groups for which we have no space. One result arising is that the simultaneous system p1= φ1, ... pr= φr, wherein p1, ... prare not involved in φ1, ... φr, if it satisfies the ½r(r − 1) relations [pi− φi, pj− φj] = 0, has a solution z = ψ(x1, ... xn), p1= dψ/dx1, ... pn= dψ/dxn, reducing to an arbitrary function of xr+1, ... xnonly, when x1= xº1, ... xr= xºrunder certain conditions as to developability; a generalization of the theorem for linear equations. The problem of integration of this system is, as before, to put
dz − φ1dx1− ... − φrdxr− pr+1dxr+1− ... − pndxn
into the form σ(dζ − ωr+1+ dξr+1− ... − ωndξn); and here ζ, ξr+1, ... ξn, ωr+1, ... ωnmay be taken, as before, to be principal integrals of a certain complete system of linear equations; those, namely, determining the characteristic chains.
If L be a function of t and of the 2n quantities x1, ... xn, ẋ1, ... ẋn, where ẋi, denotes dxi/dt, &c., and if in the n equations
we put pi= dL/dẋi, and so express ẋi, ... ẋnin terms of t, xi, ... xn, p1, ... pn, assuming that the determinant of the quantities d²L/dxidẋjis not zero; if, further, H denote the function of t, x1, ... xn, p1, ... pn, numerically equal to p1ẋ1+ ... + pnẋn− L, it is easyEquations of dynamics.to prove that dpi/dt = −dH/dxi, dxi/dt = dH/dpi. These so-calledcanonicalequations form part of those for the characteristic chains of the single partial equation dz/dt + H(t, x1, ... xn, dz/dx1, ..., dz/dxn) = 0, to which then the solution of the original equations for x1... xncan be reduced. It may be shown (1) that if z = ψ(t, x1, ... xn, c1, .. cn) + c be a complete integral of this equation, then pi= dψ/dxi, dψ/dci= eiare 2n equations giving the solution of the canonical equations referred to, where c1... cnand e1, ... enare arbitrary constants; (2) that if xi= Xi(t, x01, ... pºn), pi= Pi(t, xº1, ... p0n) be the principal solutions of the canonical equations for t = t0, and ω denote the result of substituting these values in p1dH/dp1+ ... + pndH/dpn− H, and Ω = ∫tt0ωdt, where, after integration, Ω is to be expressed as a function of t, x1, ... xn, xº1, ... xºn, then z = Ω + z0is a complete integral of the partial equation.
A system of differential equations is said to allow a certain continuous group of transformations (seeGroups, Theory of) when the introduction for the variables in the differential equations of the new variables given by theApplication of theory of continuous groups to formal theories.equations of the group leads, for all values of the parameters of the group, to the same differential equations in the new variables. It would be interesting to verify in examples that this is 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 very general application for the case of a simultaneous complete system of linear partial homogeneous differential equations of the first 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 transformations of the group.
It can be shown easily that sufficient conditions in order that a complete system Π1ƒ = 0 ... Πkƒ = 0, in n independent variables, should allow the infinitesimal transformation Pƒ = 0 are expressed by k equations ΠiPƒ − PΠiƒ = λi1Π1ƒ + ... + λikΠkƒ. Suppose now a complete system of n − r equations in n variables to allow a group of r infinitesimal transformations (P1f, ..., Prƒ) which has an invariant subgroup of r − 1 parameters (P1ƒ, ..., Pr-1ƒ), it being supposed that the n quantities Π1ƒ, ..., Πn-rƒ, P1ƒ, ..., Prƒ 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 determinable by a quadrature. For each of ΠiPσƒ − PσΠif is a linear function of Π1ƒ, ..., Πn-rƒ and the simultaneous system of independent equations Π1ƒ = 0, ... Πn-rƒ = 0, P1ƒ = 0, ... Pr-1ƒ = 0 is therefore a complete system, allowing the infinitesimal transformation Prƒ. This complete system of n − 1 equations has therefore one common solution ω, and Pr(ω) is a function of ω. By choosing ω suitably, we can then make Pr(ω) = 1. From this equation and the n − 1 equations Πiω = 0, Pσω= 0, we can determine ω 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 entirety 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 − 1 variables, subject to an integrable group of r − 1 parameters, and to continue this process. We give some examples of the application of the theorem. (1) If an equation of the first order y′ = ψ(x, y) allow the infinitesimal transformation ξdƒ/dx + ηdƒ/dy, the integral curves ω(x, y) = y0, wherein ω(x, y) is the solution of dƒ/dx + ψ(x, y) dƒ/dy = 0 reducing to y for x = x0, are interchanged among themselves by the infinitesimal transformation, or ω(x, y) can be chosen to make ξdω/dx + ηdω/dy = 1; this, with dω/dx + ψdω/dy = 0, determines ω as the integral of the complete differential (dy − ψdx)/(η − ψξ). 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 ξdƒ/dx + ηdƒ/dy can by change of variables (after integration) be brought to the form dƒ/dy, and all differential equations of the first order allowing this group can then be reduced to the form F(x, dy/dx) = 0. (2) In an ordinary equation of the second order y” = ψ(x, y, y′), equivalent to dy/dx = y1, dy1/dx = ψ(x, y, y1), if H, H1be the solutions for y and y1chosen to reduce to y0and yº1when x = x0, and the equations H = y, H1= y1be equivalent to ω = y0, ω1= yº1, then ω, ω1are the principal solutions of Πƒ = dƒ/dx + y1dƒ/dy + ψdƒ/dy1= 0. If the original equation allow an infinitesimal transformation whose firstextendedform (seeGroups) is Pƒ = ξdƒ/dx + ηdƒ/dy + η1dƒ/dy1, where η1δt is the increment of dy/dx when ξδt, ηδt are the increments of x, y, and is to be expressed in terms of x, y, y1, then each of Pω and Pω1must be functions of ω and ω1, or the partial differential equation Πƒ must allow the group Pƒ. 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 Πƒ and the two infinitesimal transformations are not linearly connected. It can be shown, from the fact that η1is a quadratic polynomial in y1, 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/dx² = 0. 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 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, dγ/dx, d²γ/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 linearly 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 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. (4) Similarly for the integration by quadratures of an ordinary equation yn= ψ(x, y, y1, ... yn-1) 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 oftransformations 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 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 bothConsideration of function theories of differential equations.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. Green, the 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= φ1, ..., pr= φr. 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 z1, ... zrand n independentA general existence theorem.variables x1, ... xn; let the differential coefficient of zσof highest order which enters be of order hσ, and suppose dhσzσ/ dx1hσto enter, so that the equations can be written dhσzσ/ dx1hσ= Φσ, where in the general differential coefficient of zρwhich enters in Φσ, say
dk1 + ... + knzρ/ dx1k1... dxnkn,
we have k1< hρand k1+ ... + kn≤ hρ. Let a1, ... an, b1, ... br, and bρk1 ... knbe a set of values of
x1, ... xn, z1, ... zr
and of the differential coefficients entering in Φσabout which all the functions Φ1, ... Φr, are developable. Corresponding to each dependent variable zσ, we take now a set of hσfunctions of x2, ... xn, say φσ, φσ;(1), ... ,φσh−1arbitrary save that they must be developable about a2, a3, ... an, and such that for these values of x2, ... xn, the function φρreduces to bρ, and the differential coefficient
dk2 + ... + knφρk1/ dx2k2... dxnkn
reduces to bρk1 ... kn. Then the theorem is that there exists one, and only one, set of functions z1, ... zr, of x2, ... xndevelopable about a1, ... ansatisfying the given differential equations, and such that for x1= a1we have
zσ= φσ, dzσ/ dx1= φσ(1), ... dhσ−1zσ/ dhσ−1x1= φσhσ−1.
And, moreover, if the arbitrary functions φσ, φσ(1)... contain a certain number of arbitrary variables t1, ... tm, and be developable about the values tº1, ... tºmof these variables, the solutions z1, ... zrwill contain t1, ... tm, and be developable about tº1, ... tºm.
The proof of this theorem may be given by showing that if ordinary power series in x1− a1, ... xn− an, t1− tº1, ... tm− tºmbe substituted in the equations wherein in zσthe coefficients of (x1− a1)º, x1− a1, ..., (x1− a1)hσ−1are the arbitrary functions φσ, φσ(1), ..., φσh−1, 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 (seeFunction), 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 bycontinuation; 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 of the independent variables with a differentSingular points of solutions.value of the function; belonging, as we say, to a different branch of the function; and there is even no 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 eitherfixed, being determined by the differential equations themselves, or they may bemovablewith the variation of the arbitrary constants of integration. Such difficulties arise even in establishing the reversion of an elliptic integral, in solving the equation
(dx/ds)² = (x − a1)(x − a2)(x − a3)(x − a4);
about an ordinary value the right side is developable; if we put x − a1= t1², the right side becomes developable about t1= 0; if we put x = 1/t, the right side of the changed equation is developable about t = 0; it is quite easy to show that the integral reducing to a definite value x0for a value s0is obtainable by a series in integral 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 homogeneousLinear differential equations with rational coefficients.differential equations whose importance arises from the completeness and generality with which they can be discussed. We have seen that if in the equations
dy/dx = y1, dy1/dx = y2, ..., dyn−2/dx = yn−1,dyn−1/dx = any + an−1y1+ ... + a1yn−1,
where a1, a2, ..., anare now to be taken to be rational functions of x, the value x = xº be one for which no one of these rational functions is infinite, and yº, yº1, ..., yºn−1be quite arbitrary finite values, then the equations are satisfied by
y = yºu + yº1u1+ ... + yºn−1un−1,
where u, u1, ..., un−1are functions of x, independent of yº, ... yºn−1, developable about x = xº; this value of y is such that for x = xº the functions y, y1... yn−1reduce respectively to yº, yº1, ... yºn−1; it can be proved that the region of existence of these series extends within a circle centre xº and radius equal to the distance from xº of the nearest point at which one of a1, ... anbecomes infinite. Now consider a region enclosing xº and only one of the places, say Σ, at which one of a1, ... anbecomes infinite. When x is made to describe a closed curve in this region, including this point Σ in its interior, it may well happen that the continuations of the functions u, u1, ..., un−1give, when we have returned to the point x, values v, v1, ..., vn−1, so that the integral under consideration becomes changed to yº + yº1v1+ ... + yºn−1vn−1. At xº let this branch and the corresponding values of y1, ... yn−1be ηº, ηº1, ... ηºn−1; then, as there is only one series satisfying the equation and reducing to (ηº, ηº1, ... ηºn−1) for x = xº and the coefficients in the differential equation are single-valued functions, we must have ηºu + ηº1u1+ ... + ηºn−1un−1= yºv + yº1v1+ ... + yºn−1vn−1; as this holds for arbitrary values of yº ... yºn−1, upon which u, ... un−1and v, ... vn−1do not depend, it follows that each of v, ... vn−1is a linear function of u, ... un−1with constant coefficients, say vi= Ai1u + ... + Ainun−1. Then
yºv + ... + yºn−1vn−1= (ΣiAi1yºi)u + ... + (ΣiAinyºi) un−1;
this is equal to μ(yºu + ... + yºn−1un−1) if ΣiAiryºi= μyºr−1; eliminating yº ... yºn−1from these linear equations, we have a determinantal equation of order n for μ; let μ1be one of its roots; determining the ratios of yº, y1º, ... yºn−1to satisfy the linear equations, we have thus proved that there exists an integral, H, of the equation, which when continued round the point Σ and back to the starting-point, becomes changed to H1= μ1H. Let now ξ be the value of x at Σ and r1one of the values of (½πi) log μ1; consider the function (x − ξ)−r1H; when x makes a circuit round x = ξ, this becomes changed to
exp (-2πir1) (x − ξ)−r1μH,
that is, is unchanged; thus we may put H = (x − ξ)r1φ1, φ1being a function single-valued for paths in the region considered described about Σ, and therefore, by Laurent’s Theorem (seeFunction), capable of expression in the annular region about this point by a series of positive and negative integral powers of x − ξ, which in general may contain an infinite number of negative powers; there is, however, no reason to suppose r1to be an integer, or even real. Thus, if all the roots of the determinantal equation in μ are different, we obtain n integrals of the forms (x − ξ)r1φ1, ..., (x − ξ)rnφn. 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 root be k times repeated, k − 1 equations of as simple a form as possible to replace the k − 1 equations of the form yº + ... + yºn−1vn−1= μ(yº + ... + yºn−1un−1) which would have existed had the roots been different. The most natural method of obtaining a suggestion lies probably in remarking that if r2= r1+ h, there is an integral [(x − ξ)r1 + hφ2− (x − ξ)r1φ1] / h, where the coefficients in φ2arethe same functions of r1+ h as are the coefficients in φ1of r1; when h vanishes, this integral takes the form
(x − ξ)r1[dφ1/dr1+ φ1log (x − ξ)],
or say
(x − ξ)r1[φ1+ ψ1log (x − ξ)];
denoting this by 2πiμ1K, and (x − ξ)r1φ1by H, a circuit of the point ξ changes K into
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 = ξ being H1′ = μ1H1, H2′ = μ1H2+ H1, H3′ = μ1H3+ H2, and so on. And this is found to lead to the forms (x − ξ)r1φ1, (x − ξ)r1[ψ1+ φ1log (x − ξ)], (x − ξ)r1[χ1+ χ2log (x − ξ) + φ1(log(x − ξ) )²], and so on. Here each of φ1, ψ1, χ1, χ2, ... is a series of positive and negative integral powers of x − ξ 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 a1, ... an, it may be possible to ensure that in each of the series φ1, ψ1, [chi]1, ... the number of negative powers shall be finite. HereinRegular equations.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 φ1, ψ1, χ1... 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
y′n= (x − ξ)−1b1y′(n−1)+ (x − ξ)−2b2y′(n−2)+ ... + (x − ξ)−nbny,
where b1, ... bnare finite for x = ξ. Conversely, assume the equation to have this form. Then on substituting a series of the form (x − ξ)r[1 + A1(x − ξ) + A2(x − ξ)² + ... ] and equating the coefficients of like powers of x − ξ, 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 − ξ)rand replacing each of b1, ... bnby their values at x = ξ; arrange the roots r1, r2, ... of this equation so that the real part of riis equal to, or greater than, the real part of ri+1, and take r equal to r1; it is found that the coefficients A1, A2... are uniquely determinate, and that the series converges within a circle about x = ξ which includes no other of the points at which the rational functions a1... anbecome infinite. We have thus a solution H1= (x − ξ)r1φ1of the differential equation. If we now substitute in the equation y = H1∫ηdx, it is found to reduce to an equation of order n − 1 for η of the form
η′(n−1)= (x − ξ)−1c1η′(n−2)+ ... + (x − ξ)(n−1)cn−1η,
where c1, ... cn−1are not infinite at x = ξ. To this equation precisely similar reasoning can then be applied; its index equation has in fact the roots r2− r1− 1, ..., rn− r1− 1; if r2− r1be zero, the integral (x − ξ)−1ψ1of the η equation will give an integral of the original equation containing log (x − ξ); if r2− r1be an integer, and therefore a negative integer, the same will be true, unless in ψ1the term in (x − ξ)r1 − r2be absent; if neither of these arise, the original equation will have an integral (x − ξ)r2φ2. The η equation can now, by means of the one integral of it belonging to the index r2− r1− 1, 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 isregularabout x = ξ.
We may examine in this way the behaviour of the integrals at all the points at which any one of the rational functions a1... anbecomes infinite; in general we must expect that beside these the value x = ∞ will be a singular point for theFuchsian equations.solutions of the differential equation. To test this we put x = 1/t throughout, and examine as before at t = 0. For instance, the ordinary linear equation with constant coefficients has no singular point for finite values of x; at x = ∞ it has a singular point and is not regular; or again, Bessel’s equation x²y″ + xy′ + (x² − n²)y = 0 is regular about x = 0, but not about x = ∞. An equation regular at all the finite singularities and also at x = ∞ 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 = 1/t, it becomes
d²y/dt² + (2t−1− at−2) dy/dt + bt−4y = 0,
which is not regular about t = 0 unless 2 − at−1and bt−2, that is, unless ax and bx² are finite at x = ∞; which we thus assume; putting y = tr(1 + A1t + ... ), we find for the index equation at x = ∞ the equation r(r − 1) + r(2 − ax)0+ (bx²)0= 0. If there beEquation of the second order.finite singular points at ξ1, ... ξm, where we assume m > 1, the cases m = 0, m = 1 being easily dealt with, and if φ(x) = (x − ξ1) ... (x − ξm), we must have a·φ(x) and b·[φ(x)]² finite for all finite values of x, equal say to the respective polynomials ψ(x) and θ(x), of which by the conditions at x = ∞ the highest respective orders possible are m − 1 and 2(m − 1). The index equation at x = ξ1is r(r − 1) + rψ(ξ1) / φ′ (ξ1) + θ(ξ)1/ [φ′(ξ1)]² = 0, and if α1, β1be its roots, we have α1+ β1= 1 − ψ(ξ1) / φ′ (ξ1) and α1β1= θ(ξ)1/ [φ′(ξ1)]². Thus by an elementary theorem of algebra, the sum Σ(1 − αi− βi) / (x − ξi), extended to the m finite singular points, is equal to ψ(x) / φ(x), and the sum Σ(1 − αi− βi) is equal to the ratio of the coefficients of the highest powers of x in ψ(x) and φ(x), and therefore equal to 1 + α + β, where α, β are the indices at x = ∞. Further, if (x, 1)m−2denote the integral part of the quotient θ(x) / φ(x), we have Σ αiβiφ′ (ξi) / (x = ξi) equal to −(x, 1)m−2+ θ(x)/φ(x), and the coefficient of xm−2in (x, 1)m−2is αβ. Thus the differential equation has the form
y″ + y′Σ (1 − αi− βi) / (x − ξi) + y[(x, 1)m-2+ Σ αiβiφ′(ξi) / (x − ξi)]/φ(x) = 0.
If, however, we make a change in the dependent variable, putting y = (x − ξ1)α1... (x − ξm)α mη, 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 = ξibecome 0 and βi− αi, which we shall denote by λi, for (x − ξi)αjcan be developed in positive integral powers of x − ξiabout x = ξi; by this transformation the indices at x = ∞ are changed to
α + α1+ ... + αm, β + β1+ ... + βm
which we shall denote by λ, μ. If we suppose this change to have been introduced, and still denote the independent variable by y, the equation has the form
y″ + y′Σ (1 − λi) / (x − ξi) + y(x, 1)m−2/ φ(x) = 0,
while λ + μ + λ1+ ... + λm= m − 1. Conversely, it is easy to verify that if λμ be the coefficient of xm−2in (x, 1)m−2, this equation has the specified singular points and indices whatever be the other coefficients in (x, 1)m−2.
Thus we see that (beside the cases m = 0, m = 1) the “Fuchsian equation” of the second order withtwofinite singular points is distinguished by the fact that it has a definite form when the singular points and the indices are assigned.Hypergeometric equation.In that case, putting (x − ξ1) / (x − ξ2) = t / (t − 1), the singular points are transformed to 0, 1, ∞, and, as is clear, without change of indices. Still denoting the independent variable by x, the equation then has the form
x(1 − x)y″ + y′[1 − λ1− x(1 + λ + μ)] − λμy = 0,
which is the ordinary hypergeometric equation. Provided none of λ1, λ2, λ − μ be zero or integral about x = 0, it has the solutions
F(λ, μ, 1 − λ1, x), xλ1F(λ + λ1, μ + λ1, 1 + λ1, x);
about x = 1 it has the solutions
F(λ, μ, 1 − λ2, 1 − x), (1 − x)λ2F(λ + λ2, μ + λ2, 1 + λ2, 1 − x),
where λ + μ + λ1+ λ2= 1; about x = ∞ it has the solutions
x−λF(λ, λ + λ1, λ − μ + 1, x−1), x−μF(μ, μ + λ1, μ − λ + 1, x−1),
where F(α, β, γ, x) is the series
which converges when |x| < 1, whatever α, β, γ may be, converges for all values of x for which |x| = 1 provided the real part of γ − α − β < 0 algebraically, and converges for all these values except x = 1 provided the real part of γ − α − β > −1 algebraically.
In accordance with our general theory, logarithms are to be expected in the solution when one of λ1, λ2, λ − μ is zero or integral. Indeed when λ1is a negative integer, not zero, the second solution about x = 0 would contain vanishing factors in the denominators of its coefficients; in case λ or μ be one of the positive integers 1, 2, ... (−λ1), vanishing factors occur also in the numerators; and then, in fact, the second solution about x = 0 becomes xλ1times an integral polynomial of degree (−λ1) − λ or of degree (−λ1) − μ. But when λ1is a negative integer including zero, and neither λ nor μ is one of the positive integers 1, 2 ... (−λ1), the second solution about x = 0 involves a term having the factor log x. When λ1is a positive integer, not zero, the second solution about x = 0 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 -λ or −μ1, when λ or μ is one of the negative integers 0, −1, −2, ..., 1 − λ1, but otherwise contains a logarithm. Similarly for the solutions about x = 1 or x = ∞; it will be seen below how the results are deducible from those for x = 0.
Denote now the solutions about x = 0 by u1, u2; those about x = 1 by v1, v2; and those about x = ∞ by w1, w2; in the region (S0S1) common to the circles S0, S1of radius 1 whose centres are the points x = 0, x = 1, all the first four are valid,March of the Integral.and there exist equations u1=Av1+ Bv2, u2= Cv1+ Dv2where A, B, C, D are constants; in the region (S1S) lying inside the circle S1and outside the circle S0, those that are valid are v1, v2, w1, w2, and there exist equations v1= Pw1+ Qw2, v2= Rw1+ Tw2, where P, Q, R, T are constants; thus considering any integral whose expression within the circle S0is au1+ bu2, where a, b are constants, the same integral will be represented within the circle S1by (aA + bC)v1+ (aB + bD)v2, and outside these circles will be represented by
[aA + bC)P + (aB + bD)R]w1+ [(aA + bC)Q + (aB + bD)T]w2.
A single-valued branch of such integral can be obtained by making a barrier in the plane joining ∞ to 0 and 1 to ∞; for instance, by excluding the consideration of real negative values of x and of realpositive values greater than 1, and defining the phase of x and x − 1 for real values between 0 and 1 as respectively 0 and π.
We can form the Fuchsian equation of the second order with three arbitrary singular points ξ1, ξ2, ξ3, and no singular point at x = ∞, and with respective indices α1, β1, α2, β2, α3, β3such that α1+ β1+ α2+ β2+ α3+ β3= 1. This equation can then beTransformation of the equation into itself.transformed into the hypergeometric equation in 24 ways; for out of ξ1, ξ2, ξ3we can in six ways choose two, say ξ1, ξ2, which are to be transformed respectively into 0 and 1, by (x − ξ1)/(x − ξ2) = t(t − 1); and then there are four possible transformations of the dependent variable which will reduce one of the indices at t = 0 to zero and one of the indices at t = 1 also to zero, namely, we may reduce either α1or β1at t = 0, and simultaneously either α2or β2at t = 1. Thus the hypergeometric equation itself can be transformed into itself in 24 ways, and from the expression F(λ, μ, 1 − λ1, x) which satisfies it follow 23 other forms of solution; they involve four series in each of the arguments, x, x − 1, 1/x, 1/(1 − x), (x − 1)/x, x/(x − 1). Five of the 23 solutions agree with the fundamental solutions already described about x = 0, x = 1, x = ∞; 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 Jacobi’s theory of elliptic functions, of which K = ∫π/20(1 − h sin ²θ)−½dθ, and K′ is the same function of 1-h, can easily be proved to be the solutions of a hypergeometricInversion. Modular functions.equation of which h is the independent variable. When K, K′ are regarded as defined in terms of h by the differential equation, the ratio K′/K is an infinitely many valued 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 (seeFunction) in terms of single-valued functions. We may then attempt to investigate, in general, in what cases the independent variable x of a hypergeometric equation is a single-valued function of the ratio s of two independent integrals of the equation. The same inquiry is suggested by the problem of ascertaining in what cases the hypergeometric series F(α, β, γ, 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 axis from -∞ to 0 and from 1 to +∞, and, supposing logarithms not to enter about x = 0, choose two quite definite integrals y1, y2of the equation, say
y1= F(λ, μ, 1 − λ1, x), y2= xλ1F(λ + λ1, μ + λ1, 1 + λ1, x),
with the condition that the phase of x is zero when x is real and between 0 and 1. Then the value of ς = y2/y1is definite for all values of x in the divided plane, ς being a single-valued monogenic branch of an analytical function existing and without singularities all over this region. If, now, the values of ς that so arise be plotted on to another plane, a value p + iq of σ being represented by a point (p, q) of this ς-plane, and the value of x from which it arose being mentally associated with this point of the σ-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 ς-plane can arise twice, or in other words, whether the region of the ς-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 -∞ to 0 be momentarily removed, and x describe a small circle with centre at x = 0 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 σ, 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 σ be similarly plotted on the ς-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/y1, 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 ς-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 = 0 from x = −h − ik to −h + ik, where h is small and k evanescent,
ς = xλ1F(λ + λ1, μ + λ1, 1 + λ1, x) / F(λ, μ, 1 − λ1, x)
is changed to σ = ςe2πiλ1. Thus the two portions of boundary of the s-region corresponding to the two sides of the barrier (−∞, 0) meet (at ς = 0 if the real part of λ1be positive) at an angle 2πL1, where L1is the absolute value of the real part of λ1; the same is true for the σ-region representing the branch σ. The condition that the s-region shall not overlap itself requires, then, L1= 1. But, further, we may form an infinite number of branches σ = ςe2πiλ1, σ1= e2πiλ1, ... in the same way, and the corresponding regions in the plane upon which y2/y1is represented will have a common point and each have an angle 2πL1; if neither overlaps the preceding, it will happen, if L1is not zero, that at length one is reached overlapping the first, unless for some positive integer α we have 2παL1= 2π, in other words L1= 1/α. If this be so, the branch σα−1= ςe2πiαλ1will be represented by a region having the angle at the common point common with the region for the branch ς; but not altogether coinciding with this last region unless λ1be real, and therefore = ±1/α; then there is only a finite number, α, of branches obtainable in this way by crossing the barrier (−∞, 0). In precisely the same way, if we had begun by taking the quotient
ς′ = (x − 1)λ2F(λ + λ2, μ + λ2, 1 + λ2, 1 − x) / F(λ, μ, 1 − λ2, 1 − x)
of the two solutions about x = 1, we should have found that x is not a single-valued function of ς′ unless λ2is the inverse of an integer, or is zero; as ς′ is of the form (Aσ+ B)/(Cς+ D), A, B, C, D constants, the same is true in our case; equally, by considering the integrals about x = ∞ we find, as a third condition necessary in order that x may be a single-valued function of ς, that λ − μ must be the inverse of an integer or be zero. These three differences of the indices, namely, λ1, λ2, λ − μ, are the quantities which enter in the differential equation satisfied by x as a function of ς, which is easily found to be
where x1= dx/dς, &c.; and h1= 1 − y1², h2= 1 − λ2², h3= 1 − (λ − μ)². Into the converse question whether the three conditions are sufficient to ensure (1) that the σ 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. (SeeGroups, 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 Galois’s theory of algebraic equations. We deal only with the application to homogeneous linear differential equations.
In general a function of variables x1, x2... is said to be rational when it can be formed from them and the integers 1, 2, 3, ... by a finite number of additions, subtractions, multiplications and divisions. We generalize this definition. Assume thatRationality group of a linear equation.we have assigned a fundamental series of quantities and functions of x, in which x itself is included, such that all quantities formed by a finite number of additions, subtractions, multiplications, divisionsand 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 calledrational. By a rational function of quantities p, 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 y(k)which enters; only such equations are here discussed. Such an equation P = 0 is calledirreduciblewhen, firstly, being arranged as an integral polynomial in y(k), this polynomialIrreducibility of a rational equation.is not the product of other polynomials in y(k)also of rational form; and, secondly, the equation has no solution satisfying also a rational equation of lower order. From this it follows that if an irreducible equation P = 0 have one solution satisfying another rational equation Q = 0 of the same or higher order, then all the solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we can by differentiation express y(k+1), y(k+2), ... in terms of x, y, y(1), ... , y(k), 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 = 0 is of the same order as P = 0. Let both the equations be arranged as integral polynomials in y(k); their algebraic eliminant in regard to y(k)must then vanish identically, for they are known to have one common solution not satisfying an equation of lower order; thus the equation P = 0 involves Q = 0 for all solutions of P = 0.
Now let y(n)= a1y(n−1)+ ... + any be a given rational homogeneous linear differential equation; let y1, ... ynbe n particular functions of x, unconnected by any equation with constant coefficients of the form c1y1+ ... + cnyn= 0, all satisfyingThe variant function for a linear equation.the differential equation; let η1, ... ηnbe linear functions of y1, ... yn, say ηi= Ai1y1+ ... + Ainyn, where the constant coefficients Aijhave a non-vanishing determinant; write (η) = A(y), these being the equations of a general linear homogeneous group whose transformations may be denoted by A, B, .... We desire to form a rational function φ(η), or say φ(A(y)), of η1, ... η, in which the η² constants Aijshall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the y1, ... ynwere connected by a linear equation with constant coefficients. Such a function is in fact given, if the solutions y1, ... ynbe developablein positive integral powers about x = a, by φ(η) = η1+ (x − a)nη2+ ... + (x − a)(n−1)nηn. Such a function, V, we call avariant.