Page:EB1911 - Volume 08.djvu/253

 the same functions of r1 + h as are the coefficients in &phi;1 of r1; when h vanishes, this integral takes the form (x &minus; &xi;)r1 [d&phi;1/dr1 + &phi;1 log (x &minus; &xi;)], or say (x &minus; &xi;)r1 [&phi;1 + &psi;1 log (x &minus; &xi;)]; denoting this by 2&pi;i&mu;1K, and (x &minus; &xi;)r1 &phi;1 by H, a circuit of the point &xi; changes K into $\mathrm{K} = \frac{1}{2\pi i\mu_{1}}\left[e^{2\pi ir_1}(x - \xi )^{r_1}\psi_{1}+ e^{2\pi ir _1}(x - \xi )^{r_1}\phi_{1}(2\pi i + \log (x - \xi) )\right] = \mu_{1}\mathrm{K}+ \mathrm{H}.$|undefined 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 = &xi; being H1′ = &mu;1H1, H2′ = &mu;1H2 + H1, H3′ = &mu;1H3 + H2, and so on. And this is found to lead to the forms (x &minus; &xi;)r1&phi;1, (x &minus; &xi;)r1 [&psi;1 + &phi;1 log (x &minus; &xi;)], (x &minus; &xi;)r1 [&chi;1 + &chi;2 log (x &minus; &xi;) + &phi;1(log(x &minus; &xi;) )²], and so on. Here each of &phi;1, &psi;1, &chi;1, &chi;2, ... is a series of positive and negative integral powers of x &minus; &xi; 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 &phi;1, &psi;1, [&chi;]1, ... the number of negative powers shall be finite. Herein 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 &phi;1, &psi;1, &chi;1 ... 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 &minus; &xi;)&minus;1 b1y′ (n&minus;1) + (x &minus; &xi;)&minus;2 b2y′ (n&minus;2) + ... + (x &minus; &xi;)&minus;n bny, where b1, ... bn are finite for x = &xi;. Conversely, assume the equation to have this form. Then on substituting a series of the form (x &minus; &xi;)r [1 + A1(x &minus; &xi;) + A2(x &minus; &xi;)² + ... ] and equating the coefficients of like powers of x &minus; &xi;, 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 &minus; &xi;)r and replacing each of b1, ... bn by their values at x = &xi;; 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 = &xi; which includes no other of the points at which the rational functions a1 ... an become infinite. We have thus a solution H1 = (x &minus; &xi;)r1&phi;1 of the differential equation. If we now substitute in the equation y = H1&int;&eta;dx, it is found to reduce to an equation of order n &minus; 1 for &eta; of the form &eta;′ (n&minus;1) = (x &minus; &xi;)&minus;1 c1&eta;′ (n&minus;2) + ... + (x &minus; &xi;)(n&minus;1) cn&minus;1&eta;, where c1, ... cn&minus;1 are not infinite at x = &xi;. To this equation precisely similar reasoning can then be applied; its index equation has in fact the roots r2 &minus; r1 &minus; 1, ..., rn &minus; r1 &minus; 1; if r2 &minus; r1 be zero, the integral (x &minus; &xi;)&minus;1&psi;1 of the &eta; equation will give an integral of the original equation containing log (x &minus; &xi;); if r2 &minus; r1 be an integer, and therefore a negative integer, the same will be true, unless in &psi;1 the term in (x &minus; &xi;)r1 &minus; r2 be absent; if neither of these arise, the original equation will have an integral (x &minus; &xi;)r2&phi;2. The &eta; equation can now, by means of the one integral of it belonging to the index r2 &minus; r1 &minus; 1, be similarly reduced to one of order n &minus; 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 = &xi;.

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 = &infin; will be a singular point for the 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 = &infin; it has a singular point and is not regular; or again, Bessel’s equation x²y″ + xy′ + (x² &minus; n²)y = 0 is regular about x = 0, but not about x = &infin;. An equation regular at all the finite singularities and also at x = &infin; 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&minus;1 &minus; at&minus;2) dy/dt + bt&minus;4 y = 0, which is not regular about t = 0 unless 2 &minus; at&minus;1 and bt&minus;2, that is, unless ax and bx² are finite at x = &infin;; which we thus assume; putting y = tr(1 + A1t + ... ), we find for the index equation at x = &infin; the equation r(r &minus; 1) + r(2 &minus; ax)0 + (bx²)0 = 0. If there be finite singular points at &xi;1, ... &xi;m, where we assume m &gt; 1, the cases m = 0, m = 1 being easily dealt with, and if &phi;(x) = (x &minus; &xi;1) ... (x &minus; &xi;m), we must have a·&phi;(x) and b·[&phi;(x)]² finite for all finite values of x, equal say to the respective polynomials &psi;(x) and &theta;(x), of which by the conditions at x = &infin; the highest respective orders possible are m &minus; 1 and 2(m &minus; 1). The index equation at x = &xi;1 is r(r &minus; 1) + r&psi;(&xi;1) / &phi;′ (&xi;1) + &theta;(&xi;)1 / [&phi;′(&xi;1)]² = 0, and if &alpha;1, &beta;1 be its roots, we have &alpha;1 + &beta;1 = 1 &minus; &psi;(&xi;1) / &phi;′ (&xi;1) and &alpha;1&beta;1 = &theta;(&xi;)1 / [&phi;′(&xi;1)]². Thus by an elementary theorem of algebra, the sum &Sigma;(1 &minus; &alpha;i &minus; &beta;i) / (x &minus; &xi;i), extended to the m finite singular points, is equal to &psi;(x) / &phi;(x), and the sum &Sigma;(1 &minus; &alpha;i &minus; &beta;i) is equal to the ratio of the coefficients of the highest powers of x in &psi;(x) and &phi;(x), and therefore equal to 1 + &alpha; + &beta;, where &alpha;, &beta; are the indices at x = &infin;. Further, if (x, 1)m&minus;2 denote the integral part of the quotient &theta;(x) / &phi;(x), we have &Sigma; &alpha;i&beta;i&phi;′ (&xi;i) / (x = &xi;i) equal to &minus;(x, 1)m&minus;2 + &theta;(x)/&phi;(x), and the coefficient of xm&minus;2 in (x, 1)m&minus;2 is &alpha;&beta;. Thus the differential equation has the form y″ + y′&Sigma; (1 &minus; &alpha;i &minus; &beta;i) / (x &minus; &xi;i) + y[(x, 1)m-2 + &Sigma; &alpha;i&beta;i&phi;′(&xi;i) / (x &minus; &xi;i)]/&phi;(x) = 0. If, however, we make a change in the dependent variable, putting y = (x &minus; &xi;1)&alpha;1 ... (x &minus; &xi;m)&alpha; m&eta;, 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 = &xi;i become 0 and &beta;i &minus; &alpha;i, which we shall denote by &lambda;i, for (x &minus; &xi;i)&alpha;j can be developed in positive integral powers of x &minus; &xi;i about x = &xi;i; by this transformation the indices at x = &infin; are changed to &alpha; + &alpha;1 + ... + &alpha;m, &beta; + &beta;1 + ... + &beta;m which we shall denote by &lambda;, &mu;. If we suppose this change to have been introduced, and still denote the independent variable by y, the equation has the form y″ + y′&Sigma; (1 &minus; &lambda;i) / (x &minus; &xi;i) + y(x, 1)m&minus;2 / &phi;(x) = 0, while &lambda; + &mu; + &lambda;1 + ...  + &lambda;m = m &minus; 1. Conversely, it is easy to verify that if &lambda;&mu; be the coefficient of xm&minus;2 in (x, 1)m&minus;2, this equation has the specified singular points and indices whatever be the other coefficients in (x, 1)m&minus;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. In that case, putting (x &minus; &xi;1) / (x &minus; &xi;2) = t / (t &minus; 1), the singular points are transformed to 0, 1, &infin;, and, as is clear, without change of indices. Still denoting the independent variable by x, the equation then has the form x(1 &minus; x)y″ + y′[1 &minus; &lambda;1 &minus; x(1 + &lambda; + &mu;)] &minus; &lambda;&mu;y = 0, which is the ordinary hypergeometric equation. Provided none of &lambda;1, &lambda;2, &lambda; &minus; &mu; be zero or integral about x = 0, it has the solutions F(&lambda;, &mu;, 1 &minus; &lambda;1, x), x&lambda;1 F(&lambda; + &lambda;1, &mu; + &lambda;1, 1 + &lambda;1, x); about x = 1 it has the solutions F(&lambda;, &mu;, 1 &minus; &lambda;2, 1 &minus; x), (1 &minus; x)&lambda;2 F(&lambda; + &lambda;2, &mu; + &lambda;2, 1 + &lambda;2, 1 &minus; x), where &lambda; + &mu; + &lambda;1 + &lambda;2 = 1; about x = &infin; it has the solutions x&minus;&lambda; F(&lambda;, &lambda; + &lambda;1, &lambda; &minus; &mu; + 1, x&minus;1), x&minus;&mu; F(&mu;, &mu; + &lambda;1, &mu; &minus; &lambda; + 1, x&minus;1), where F(&alpha;, &beta;, &gamma;, 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| &lt; 1, whatever &alpha;, &beta;, &gamma; may be, converges for all values of x for which |x| = 1 provided the real part of &gamma; &minus; &alpha; &minus; &beta; &lt; 0 algebraically, and converges for all these values except x = 1 provided the real part of &gamma; &minus; &alpha; &minus; &beta; &gt; &minus;1 algebraically.

In accordance with our general theory, logarithms are to be expected in the solution when one of &lambda;1, &lambda;2, &lambda; &minus; &mu; is zero or integral. Indeed when &lambda;1 is a negative integer, not zero, the second solution about x = 0 would contain vanishing factors in the denominators of its coefficients; in case &lambda; or &mu; be one of the positive integers 1, 2, ... (&minus;&lambda;1), vanishing factors occur also in the numerators; and then, in fact, the second solution about x = 0 becomes x&lambda;1 times an integral polynomial of degree (&minus;&lambda;1) &minus; &lambda; or of degree (&minus;&lambda;1) &minus; &mu;. But when &lambda;1 is a negative integer including zero, and neither &lambda; nor &mu; is one of the positive integers 1, 2 ... (&minus;&lambda;1), the second solution about x = 0 involves a term having the factor log x. When &lambda;1 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 -&lambda; or &minus;&mu;1, when &lambda; or &mu; is one of the negative integers 0, &minus;1, &minus;2, ..., 1 &minus; &lambda;1, but otherwise contains a logarithm. Similarly for the solutions about x = 1 or x = &infin;; 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 = &infin; 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, 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 &infin; to 0 and 1 to &infin;; for instance, by excluding the consideration of real negative values of x and of real