Page:EB1911 - Volume 08.djvu/253

the same functions of r₁ + h as are the coefficients in φ₁ of r₁; when h vanishes, this integral takes the form

(x − ξ)^r1 [dφ₁/dr₁ + φ₁ log (x − ξ)],

or say

(x − ξ)^r₁ [φ₁ + ψ₁ log (x − ξ)];

denoting this by 2πiμ₁K, and (x − ξ)^r₁ φ₁ by H, a circuit of the point ξ changes K into

$\mathrm {K} {=}{\frac {1}{2\pi i\mu _{1}}}\left[e^{2\pi ir_{1}}(x-\xi )^{r_{1}}\psi _{1}+e^{2\pi ir_{1}}(x-\xi )^{r_{1}}\phi _{1}(2\pi i+\log(x-\xi ))\right]{=}\mu _{1}\mathrm {K} +\mathrm {H} .$

A similar artifice suggests itself when three of the roots of the determinantal equation are the same, and so on. We are thus led to the result, which is justified by an examination of the algebraic conditions, that whatever may be the circumstances as to the roots of the determinantal equation, n integrals exist, breaking up into batches, the values of the constituents H₁, H₂, ... of a batch after circuit about x = 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 + φ₁ log (x − ξ)], (x − ξ)^r₁ [χ₁ + χ₂ log (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, ... a_n, it may be possible to ensure that in each of the series φ₁, ψ₁, [χ]₁, ... the number of negative powers shall be finite. Herein Regular 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)⁻¹ b₁y′ ⁽ⁿ⁻¹⁾ + (x − ξ)⁻² b₂y′ ⁽ⁿ⁻²⁾ + ... + (x − ξ)⁻ⁿ b_ny,

where b₁, ... b_n are 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)^r and replacing each of b₁, ... b_n by their values at x = ξ; arrange the roots r₁, r₂, ... of this equation so that the real part of r_i is equal to, or greater than, the real part of r_i+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 ... a_n become infinite. We have thus a solution H₁ = (x − ξ)^r1φ₁ of 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)⁻¹ c₁η′ ⁽ⁿ⁻²⁾ + ... + (x − ξ)⁽ⁿ⁻¹⁾ c_n−1η,

where c₁, ... c_n−1 are 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, ..., r_n − r₁ − 1; if r₂ − r₁ be zero, the integral (x − ξ)⁻¹ψ₁ of the η equation will give an integral of the original equation containing log (x − ξ); if r₂ − r₁ be an integer, and therefore a negative integer, the same will be true, unless in ψ₁ the term in (x − ξ)^{r₁ − r₂} be 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 is regular about 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 ... a_n becomes infinite; in general we must expect that beside these the value x = ∞ will be a singular point for the Fuchsian 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⁻⁴ y = 0,

which is not regular about t = 0 unless 2 − at⁻¹ and bt⁻², that is, unless ax and bx² are finite at x = ∞; which we thus assume; putting y = t^r(1 + A₁t + ... ), we find for the index equation at x = ∞ the equation r(r − 1) + r(2 − ax)₀ + (bx²)₀ = 0. If there be Equation of the second order. finite singular points at ξ₁, ... ξ_m, where we assume m > 1, the cases m = 0, m = 1 being easily dealt with, and if φ(x) = (x − ξ₁) ... (x − ξ_m), 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 = ξ₁ is r(r − 1) + rψ(ξ₁) / φ′ (ξ₁) + θ(ξ)₁ / [φ′(ξ₁)]² = 0, and if α₁, β₁ be 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 − α_i − β_i) / (x − ξ_i), extended to the m finite singular points, is equal to ψ(x) / φ(x), and the sum Σ(1 − α_i − β_i) 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−2 denote the integral part of the quotient θ(x) / φ(x), we have Σ α_iβ_iφ′ (ξ_i) / (x = ξ_i) equal to −(x, 1)_m−2 + θ(x)/φ(x), and the coefficient of x^m−2 in (x, 1)_m−2 is αβ. Thus the differential equation has the form

y″ + y′Σ (1 − α_i − β_i) / (x − ξ_i) + y[(x, 1)_m-2 + Σ α_iβ_iφ′(ξ_i) / (x − ξ_i)]/φ(x) = 0.

If, however, we make a change in the dependent variable, putting y = (x − ξ₁)^α1 ... (x − ξ_m)^{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 = ξ_i become 0 and β_i − α_i, which we shall denote by λ_i, for (x − ξ_i)^αj can be developed in positive integral powers of x − ξ_i about x = ξ_i; by this transformation the indices at x = ∞ are changed to

α + α₁ + ... + α_m, β + β₁ + ... + β_m

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 − λ_i) / (x − ξ_i) + y(x, 1)_m−2 / φ(x) = 0,

while λ + μ + λ₁ + ... + λ_m = m − 1. Conversely, it is easy to verify that if λμ be the coefficient of x^m−2 in (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 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. 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), x^λ1 F(λ + λ₁, μ + λ₁, 1 + λ₁, x);

about x = 1 it has the solutions

F(λ, μ, 1 − λ₂, 1 − x), (1 − x)^λ2 F(λ + λ₂, μ + λ₂, 1 + λ₂, 1 − x),

where λ + μ + λ₁ + λ₂ = 1; about x = ∞ it has the solutions

x^−λ F(λ, λ + λ₁, λ − μ + 1, x⁻¹), x^−μ F(μ, μ + λ₁, μ − λ + 1, x⁻¹),

where F(α, β, γ, x) is the series

$1+{\frac {\alpha \beta x}{\gamma }}+{\frac {\alpha (\alpha +1)\beta (\beta +1)x^{2}}{1\cdot 2\cdot \gamma (\gamma +1)}}\ldots ,$

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 λ₁ is 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 x^λ1 times an integral polynomial of degree (−λ₁) − λ or of degree (−λ₁) − μ. But when λ₁ is 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 λ₁ is 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, 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