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, S1 of 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 + Dv2 where A, B, C, D are constants; in the region (S1S) lying inside the circle S1 and 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 S0 is au1 + bu2, where a, b are constants, the same integral will be represented within the circle S1 by (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 real 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)−½dθ, and K′ is the same function of 1-h, can easily be proved to be the solutions of a hypergeometric Inversion. 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 (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 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, 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 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 = 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 s-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 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/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)