Hence we can deduce the remarkable result that the value of [f](z) at any interior point of a region is expressible in terms of the value of [f](z) at the boundary points. For consider in the original region the function [f](z)/(z - z0), where z0 is an interior point: this satisfies the same conditions as [f](z) except in the immediate neighbourhood of z0. Taking out then from the original region a small regular polygonal region with z0 as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral [int] dz[f](z)/(z - z0) round the boundary of the original region is equal to the same integral taken counter-clockwise round a small circle having z0 as centre; on this circle, however, if z - z0 = rE(i[theta]), dz/(z - z0) = id[theta], and [f](z) differs arbitrarily little from f(z0) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to 2[pi]i[f](z0). Hence [f](z0) = 1 / 2[pi]i [int] [dt[f](t)/(t - z0)], where this integral is round the boundary of the original region. From this it appears that _ [f](z) - [f](z0) 1 / dt[f](t) F(z0) = lim. ---------------- = ------ | -------- z - z0 2[pi]i _/ (t-z0)²
also round the boundary of the original region. This form shows, however, that F(z0) is a continuous, finite, differentiable function of z0 over the whole interior of the original region.
§ 5. _Applications._--The previous results have manifold applications.
(1) If an infinite series of differentiable functions of z be uniformly convergent along a certain path lying with the region of definition of the functions, so that S(2) = u0(z) + u1(z) + ... + u_(n - 1)(z) + R_n(z), where |R_n(z)| < [epsilon] for all points of the path, we have _ _ _ _ _ /z /z /z /z /z | S(z)dz = | u0(z)dz + | u1(z)dz + ... + | u_(n - 1)(z)dz + | R_n(z)dz, _/z0 _/z0 _/z0 _/z0 _/z0
wherein, in absolute value, [int](z_0 to z) R_n(z)dz < [epsilon]L, if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent.
(2) If [f](x, y) be definite, finite and continuous at every point of a region, and over any closed path in the region [int][f](x, y)dz = 0, then [psi](z) = [int](z_0 to z) [f](x, y)dz, for interior points z0, z, is a differentiable function of z, having for its differential coefficient the function [f](x, y), which is therefore also a differentiable function of z at interior points.