Hence it can be proved that, for a differentiable function ƒ(z), the integral ∫ zz1 ƒ(z)dz has the same value by whatever path within the region we pass from z1 to z. This we prove by showing that when taken round a closed path in the region the integral ∫ƒ(z)dz vanishes. Consider first a triangle over which the condition (z, z0) holds, for some position of z0 and every position of z, within or upon the boundary of the triangle. Then as
ƒ(z) = ƒ(z0) + (z − z0) F(z0) + ηθ(z − z0), where |θ| < 1,
we have
∫ƒ(z)dz = [ƒ(z0) − z0 F(z0)] ∫dz + F(z0) ∫zdz + η∫θ(z − z0)dz,
which, as the path is closed, is η ∫θ(z − z0)dz. Now, from the theorem that the absolute value of a sum is less than the sum of the absolute values of the terms, this last is less, in absolute value, than ηap, where a is the greatest side of the triangle and p is its perimeter; if Δ be the area of the triangle, we have Δ = ½ab sin C > (α/π) ba, where α is the least angle of the triangle, and hence a(a + b + c) < 2a(b + c) < 4πΔ/α; the integral ∫ƒ(z)dz round the perimeter of the triangle is thus < 4πηΔ/α. Now consider any region made up of triangles, as before explained, in each of which the condition (z, z0) holds, as in the triangle just taken. The integral ∫ƒ(z)dz round the boundary of the region is equal to the sum of the values of the integral round the component triangles, and thus less in absolute value than 4πηK/α, where K is the whole area of the region, and α is the smallest angle of the component triangles. However small η be taken, such a division of the region into a finite number of component triangles has been shown possible; the integral round the perimeter of the region is thus arbitrarily small. Thus it is actually zero, which it was desired to prove. Two remarks should be added: (1) The theorem is proved only on condition that the closed path of integration belongs to the region at every point of which the conditions are satisfied. (2) The theorem, though proved only when the region consists of triangles, holds also when the boundary points of the region consist of one or more closed paths, no two of which meet.
Hence we can deduce the remarkable result that the value of ƒ(z) at any interior point of a region is expressible in terms of the value of ƒ(z) at the boundary points. For consider in the original region the function ƒ(z)/(z − z0), where z0 is an interior point: this satisfies the same conditions as ƒ(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 ∫ dzƒ(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(iθ), dz/(z − z0) = idθ, and ƒ(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 2πiƒ(z0). Hence ƒ(z0) = 1/2πi ∫ (dtƒ(t)/(t − z0), where this integral is round the boundary of the original region. From this it appears that
| F(z0) = lim. | ƒ(z) − ƒ(z0) | = | 1 | ∫ | dtƒ(t) |
| z − z0 | 2πi | (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) + ... + un−1(z) + Rn(z), where |Rn(z)| < ε for all points of the path, we have