Page:Proceedings of the Royal Society of London Vol 69.djvu/130

122 to the dependence of a. n upon n. When a n is, for n large, to the first approximation a transcendental function of n, the integral function is said to be one of transcendental sequence.

A function with a finite number of simple sequences of zeros can be built up of a number of non-repeated functions each of simple sequence,

When the nth zero of a function of simple sequence is repeated a number of times dependent upon n, we call it a repeated integral function.

The typical zero may require more than a single number to define its position in the series of zeros to which it belongs ; the integral function is then called one of multiple sequence.

Finally, it is pointed out that the category to which a typical zero belongs may be indefinitely complex, and that, therefore, we can expect to lay down no general law relating to all integral functions which is not a disguised truism. In the memoir we confine ourselves substantially to simple functions with repeated or non-repeated sequences of zeros.

Part II contains the theory of divergent arid asymptotic series, Poincare's arithmetic definition is first given ; the function J (i) admits the asymptotic expansion

in certain regions where \z\ is very large, if the sum of the first n terms be s n, and the expression

tends to zero as s tends to infinity in those regions.

The difficulty of this theory is pointed out, and it is shown that a theory dependent on analytic functionality is more tractable and no less precise.

A series

of finite radius of convergence can, by an extension of a method due to Borel, be interpreted by means of a contour integral for all values of the variable outside the circle of convergence, except those which lie along the straight line joining the singularities to infinity, and proceeding directly away from the origin, provided all the singularities of the function represented by the series lie on such a line. The integral is called the " sum " of the divergent series. The process of summation is not unique, but must always lead to the same result, namely, to the function which is the analytic continuation of the function represented by the given series when convergent. On the