The general formula then gives

bn=-[1+0(n-1)] exp [(1+k) pnk — § (1+k) kpnk − 1 − } π2 (k − 1 ) k − 1 (1+k)-1p-1n-k].

It is worth noticing that this formula is independent of σ, from which fact it follows that the nth zero of 2F" (2), and therefore the (n-1)th zero b'n-1 of F" (z), is given by the same formula as bn. It is easily shown that the distance between b, and b2-1 (or b2+1) is equal to |b|nk-1, multiplied by a factor finite both ways, while the distance between b1 and b'n-1 is equal to |bn|n−1, multiplied by a similar factor. It thus appears that the large zeros of F" (2) lie relatively near to those of F(2), and do not, for example, lie approximately half-way between consecutive zeros of F' (z).

§ 58. It was seen in § 55 that the annuli An, which together cover the z-plane completely, each contain exactly one zero of F(2) when n is large. This result has an immediate consequence of some interest. If all the coefficients cn are real, the imaginary zeros of F(z) occur in conjugate pairs. But if A contains one imaginary point it evidently contains its conjugate. Consequently, after a certain value of n, the zero contained in A, must be real. If, therefore, the coefficients of a function F(z), satisfying the conditions of § 38, are real, F(z) can have only a finite number of imaginary zeros

n

OF THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY

VOLUME XXI. No. XIII. pp. 361-376.

ON A CHANGE OF ORDER OF INTEGRATION IN AN
IMPROPER REPEATED INTEGRAL.

BY

W. H. YOUNG, Sc. D., F.R.S.

CAMBRIDGE:

AT THE UNIVERSITY PRESS

M.DCCCC. X.

ADVERTISEMENT

THE Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors.

THE SOCIETY takes this opportunity of expressing its grateful acknowledgments to the SYNDICS of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions.

XIII. On the Change of Order of Integration in an Improper

Repeated Integral. By W. H. YOUNG, Sc.D., F.R.S.

[Received Dec. 1, 1909. Read 21 Feb. 1910.]

§ 1. THE object of the present paper is to give a set of rules for determining when the process of reversing the order of integration in a repeated integral, which for simplicity will be supposed to be in two variables, is allowable. It will be found that the account differs both in form and substance from others which have been hitherto presented. As regards form, no use is made, either in the enunciations of theorems or in the applications to examples, of e-machinery. In this subject, even more than in others, the use of it appears to obscure the issue. As regards substance, certain new rules are given, which, though obtained without difficulty, and applied with considerable ease, have apparently not been stated.

De la Vallée Poussin was one of the first to occupy himself with this subject. His conditions-exception being made of one far-reaching theorem due to him, which is capable of remarkable generalisation-involve considerations of uniform and non-uniform convergence. The recent trend of research, especially where integration is involved, has been materially to reduce the importance of these concepts as compared with that of being bounded. In the present paper the concepts of uniform and non-uniform convergence are not employed, but the rules given include none the less those of de la Vallée Poussin as a particular case.

I have attempted to make the exposition as systematic as possible, and have accordingly begun by stating the known result for bounded integrands, and then have proceeded successively to unbounded integrands and infinite domains.

§ 2. We begin then by considering the integral ["dy ["f(x, y) da, where p and q are finite

0

and positive and where f(x, y) is a bounded function of (x, y), and not merely of x and y separately. Exception being made of what we may call at most philosophically possible but mathematically non-existent functions, f(x, y) has then necessarily a Lebesgue double integral, and therefore its repeated Lebesgue integrals necessarily exist and are equal. Thus the change of order of Lebesgue integration is always allowable. It will however usually happen that f(x, y) possesses an ordinary Riemann integral with respect to each of the variables separately. VOL. XXI. No. XIII.

48