positive 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, β3 such that α1+β1+α2+β2+α3+β3＝1. This equation can then be Transformation of the equation into itself.transformed into the hypergeometric equation in 24 ways; for out of ξ1, ξ2, ξ3 we 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 α1 or β1 at t＝0, and simultaneously either α2 or β2 at 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)− 12dθ, and K′ is the same function of 1–h, can easily be proved to be the solutions of a hypergeometric equation of which h is the independent variable. When K, K′ are Inversion. Modular functions.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 (see Function) 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 ς 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, y2 of the equation, say
y1＝F(λ, μ, 1−λ1, x), y2＝xλ1 F(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/y1 is 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 ς 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 ς 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＝xλ1 F(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 ς-region corresponding to the two sides of the barrier (−∞, 0) meet (at ς＝0 if the real part of λ1 be positive) at an angle 2πL1, where L1 is 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 ς-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/y1 is represented will have a common point and each have an angle 2πL1; if neither overlaps the preceding, it will happen, if L1 is 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＝ςe2πiαλ1 will 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 λ1 be 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)λ2 F(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 λ2 is 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
| − | x111 | + | 3x211 | ＝12(h−h1−h2)x−1(x−1)−1+12h1x−2+12h2(x−1)−2, |
| x13 | 2x14 |
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. (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 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 that we have assigned a fundamental series of quantities and functions Rationality group of a linear equation.of x, in which x itself is included, such that all quantities formed by a finite number of additions, subtractions, 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 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 called irreducible when, firstly, being arranged as an integral polynomial in y(k), this polynomial Irreducibility 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, . . . yn be n particular functions of x, unconnected by any equation with constant coefficients of the form c1y1+ . . . +cnyn＝0, all satisfying the differential equation; let η1, . . . ηn be linear functions The variant function for a linear equation.of y1, . . . yn, say ηi＝Ai1y1+ . . . +Ainyn, where the constant coefficients Aij have 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 Aij shall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the y1, . . . yn were connected by a linear equation with constant coefficients. Such a function is in fact given, if the solutions y1, . . . yn be developable
