By Kmowles I.W. (ed.), Saito Y. (ed.)
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40 THE MADISON COLLOQUIUM. From this theorem it follows that, when the function
The singularities of the second kind. * In any case, a suitable homogeneous linear transformation of zi, •••, zn will yield a new function for which the statement is true; cf. IV , § 1. The theorems of the paragraph just cited are assumed in the present paragraph. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 53 A function which has no other singularities in a given region or in the neighborhood of a given point than non-essential ones is said to be meromorphic in the region or in the point. § 3. E ssential Singularities An analytic function of a single com plex variable z m ay have an isolated essential singularity, z = a, of either one of tw o kinds: (a) the function may be analytic throughout the com plete neighborhood of the point a except at the point itself, and there neither remain finite nor becom e infinite; (6) the function may have poles that cluster about the point a, being analytic at all other points of the neighborhood distinct from a.
The requirements can be stated in terms of a transformation, usually linear, though not necessarily projective, applied to the points of space proper, and the behavior of the transformed function in the neighborhood of a point or points for which the latter'function is not defined. Thus a function of a single com plex variable, f(z), can be defined as analytic at infinity without introducing any ideal element whatever if we proceed in either one of the following ways. In both cases we shall demand that / ( 2) be analytic outside of a certain circle in the 2-plane, and finite along this circle.
Differential Equations and Mathematical Physics by Kmowles I.W. (ed.), Saito Y. (ed.)