Selected Topics in the Theory of Divergent Series and of Continued Fractions, Vol. 1 (Classic Reprint)

Ei BOSTON COLLOQUIUM. circle of -convergence is -not a natural boundary. Then the series defines within this circle an analytic function. In the region of divergence without the circle the value of the function may be obtained by the familiar process of analytic continuation. The oretically the determination of the function is a satisfactory one, for Poincare has shown that the function throughout the domain in which it is regular can be obtained by means of an enumerable set of elements, Pn(x an). Practically, however, when Weierstrass process is employed for analytic continuation, the labor is so excessive as to render the process nearly valueless except for purposes of definition. Hence to-day a search is being made for a workable substitute. I may refer particularly in this connection to the investigations by Borel and Mittag-L effler. As I consider the work of the former to be both suggestive and practical, I have taken it as the basis of my second lecture. A second aspect of our topic, intimately connected with the continuation of the function defined by (1), is the determination of the position and character of its singularities in the region where the series diverges. This subject is treated in Lecture 3. When the circle of convergence is a natural boundary, it does not appear to be impossible, despite the earlier view of Poincare to the contrary,! to discover, at least in a certain class of cases, an appropriate, although a non-analytic mode of continuing the function across the boundary into other regions where it will be again analytic. The thesis of Borel and its recent continuation in the Ada Mathematica, together with some excellent remarks by Fabry, appear to be about all that has been done in this direction. A very brief discussion of the subject will be given in the fourth lecture in connection with series of polynomials and of rational frac
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