| ∫ zz0 S(z)dz = ∫ zz0 u0(z)dz + ∫ zz0 u1(z)dz + ... + ∫ zz0 un−1(z)dz + ∫ zz0 Rn(z)dz, |
wherein, in absolute value, ∫ zz0 Rn(z)dz < εL, if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent.
(2) If ƒ(x, y) be definite, finite and continuous at every point of a region, and over any closed path in the region ∫ƒ(x, y)dz = 0, then ψ(z) = ∫ zz0 ƒ(x, y)dz, for interior points z0, z, is a differentiable function of z, having for its differential coefficient the function ƒ(x, y), which is therefore also a differentiable function of z at interior points.
(3) Hence if the series u0(z) + u1(z) + ... to ∞ be uniformly convergent over a region, its terms being differentiable functions of z, then its sum S(z) is a differentiable function of z, whose differential coefficient, given by (1/2πi) ∫ 2πi/(t − z)², is obtainable by differentiating the series. This theorem, unlike (1), does not hold for functions of a real variable.
(4) If the region of definition of a differentiable function ƒ(z) include the region bounded by two concentric circles of radii r, R, with centre at the origin, and z0 be an interior point of this region,
| ƒ(z0) = | 1 | ∫ | ƒ(t)dt | − | 1 | ∫ | ƒ(t)dt | , |
| 2πi | Rt − z0 | 2πi | rt − z0 |
where the integrals are both counter-clockwise round the two circumferences respectively; putting in the first (t − z0)−1 = Σn=0 z0n/tn+1, and in the second (t − z0)−1 = − Σn=0 tn/z0n+1, we find ƒ(z0) = Σ ∞−∞ Anz0n, wherein An = (1/2πi) ∫ [ƒ(t)/tn+1] dt, taken round any circle, centre the origin, of radius intermediate between r and R. Particular cases are: (α) when the region of definition of the function includes the whole interior of the outer circle; then we may take r = 0, the coefficients An for which n < 0 all vanish, and the function ƒ(z0) is expressed for the whole interior |z0| < R by a power series Σ ∞0 Anz0n. In other words, about every interior point c of the region of definition a differentiable function of z is expressible by a power series in z − c; a very important result.
(β) If the region of definition, though not including the origin, extends to within arbitrary nearness of this on all sides, and at the same time the product zmƒ(z) has a finite limit when |z| diminishes to zero, all the coefficients An for which n < −m vanish, and we have
f(z0) = A−mz0−m + A−m+1z0−m+1 + ... + A−1z0−1 + A0 + A1z0 ... to ∞.
Such a case occurs, for instance, when ƒ(z) = cosec z, the number m being unity.