P1(z) + {P2(z) − P1(z)} + {P3(z) − P2(z)} + ...,
whose sum to n terms is Pn(z), converges for all finite values of z and represents ƒ(z) within C0.
When C consists of a series of disconnected polygons, some of which may include others, and, by increasing indefinitely the number of sides of the polygons C, the points C become the boundary points Γ of a region, we can suppose the poles of the rational function, constructed to approximate to ƒ(z) within R0, to be at points of Γ. A series of rational functions of the form
H1(z) + {H2(z) − H1(z)} + {H3(z) − H2(z)} + ...
then, as before, represents ƒ(z) within R0. And R0 may be taken to coincide as nearly as desired with the interior of the region bounded by Γ.
§ 11. Expression of (1 − z)−1 by means of Polynomials. Applications.—We pursue the ideas just cursorily explained in some further detail.
Let c be an arbitrary real positive quantity; putting the complex variable ζ = ξ + iη, enclose the points ζ = l, ζ = 1 + c by means of (i.) the straight lines η = ±a, from ξ = l to ξ = 1 + c, (ii.) a semicircle convex to ζ = 0 of equation (ξ − 1)2 + η2 = a2, (iii.) a semicircle concave to ζ = 0 of equation (ξ − 1 − c)2 + η2 = a2. The quantities c and a are to remain fixed. Take a positive integer r so that 1/r (c/a) is less than unity, and put σ = 1/r (c/a). Now take
c1 = 1 + c/r, c2 = 1 + 2c/r, ... cr = 1 + c;
if n1, n2, ... nr, be positive integers, the rational function