ABSTRACT
It was shown (Chapter 23) that a function holomorphic in a closed singly-connected region can be represented by an ascending series of integral powers. If the function has singularities in the interior of the region, they can be surrounded by an inner loop or boundary that together with the outer loop or boundary specifies a ring-shaped or doubly-connected region; within this region the function is holomorphic and hence takes extreme (maximum or minimum) values of the modulus on the boundary (Section 25.1). The function can then be expanded into a series of: (i) ascending powers, associated with the outer loop as for the Lagrange-Burmann series; (ii) descending powers, associated with the inner loop that extend the Lagrange-Burmann to the Teixeira series (Section 25.5). The Teixeira (Lagrange-Burmann) series for a singular (holomorphic) function, proceeds in ascending and descending (ascending only) powers of an auxiliary function f(z), and converge in a doubly-(singly-) connected region, of generally noncircular shape; choosing the auxiliary function f(z) = z − a leads to the Laurent (Taylor) series [Section 23.7 (25.6)], valid in an annulus (circle). Taking the center at the origin a = 0 yields the Laurent-Maclaurin (Stirling-Maclaurin) series; this completes the hierarcity of series expansions (Diagram 25.1), from the most general (Teixeira’s) is the most restricted (StirlingMaclaurin’s). The series, whether ascending or ascending-descending (Section 25.2) converge absolutely (Section 25.3) [uniformly (Section 25.4)] in the interior of the region (a closed subregion). The ascending power series for holomorphic functions are a particular case of the ascending-descending series for singular functions, obtained in the absence of singularities, when the inner loop can be shrunk to zero, so that the coefficients of descending powers vanish. The examples of series include the geometric (Section 25.2) and binomial (Section 25.9) series. It is possible to define a function of a series, for example, the series (Section 25.8) for the inverse function. The ascending-descending power series represent the potential flow near an isolated singularity, and show that: (i) in addition to the superposition of corner flows of angles submultiples of π that represent the regular part (Chapter 23); (ii) there is a singular part (Chapter 25) corresponding to a multipole expansion. These general representations of nonsingular (Chapter 23) [singular (Chapter 25)] fields also apply to higher dimensions than the plane; the latter is the simplest instance beyond the straight line, and more illuminating.
