| K′ = | 1 | [e2πir1 (x − ξ)r1 ψ1 + e2πir j (x − ξ)r1 φ1 (2πi + log(x − ξ) )] = μ1K + H. |
| 2πiμ1 |
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, ... 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. 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 bny,
where b1, ... bn 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, ... bn by their values at x = ξ; arrange the roots r1, r2, ... of this equation so that the real part of ri is 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 ... an 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) cn−1η,
where c1, ... cn−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, ..., rn − 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 ... an 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