which is equal to
Σr Σi (zr, i+1 − zr, i) (ƒr, i − ƒr),
is, when |zr+1 − zr| is small enough, to ensure |ƒ(zr+1) − ƒ(zr)| < η, less in absolute value than
Σ2η Σ |zr, i+1 − zr, i|,
which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 2ηS, and is therefore arbitrarily small.
The limit in question is called ∫ zz0 ƒ(z)dz. In particular when ƒ(z) = 1, it is obvious from the definition that its value is z − z0; when ƒ(z) = z, by taking ƒr = ½(zr+1 − zr), it is equally clear that its value is ½(z² − z0²); these results will be applied immediately.
Suppose now that to every interior and boundary point z0 of a certain region there belong two definite finite numbers ƒ(z0), F(z0), such that, whatever real positive quantity η may be, a real positive number ε exists for which the condition
| | | ƒ(z) − ƒ(z0) | − F(z0) | < η, |
| z − z0 |
which we describe as the condition (z, z0), is satisfied for every point z, within or upon the boundary of the region, satisfying the limitation |z − z0| < ε. Then ƒ(z0) is called a differentiable function of the complex variable z0 over this region, its differential coefficient being F(z0). The function ƒ(z0) is thus a continuous function of the real variables x0, y0, where z0 = x0 + iy0, over the region; it will appear that F(z0) is also continuous and in fact also a differentiable function of z0.
Supposing η to be retained the same for all points z0 of the region, and σ0 to be the upper limit of the possible values of ε for the point z0, it is to be presumed that σ0 will vary with z0, and it is not obvious as yet that the lower limit of the values of σ0 as z0 varies over the region may not be zero. We can, however, show that the region can be divided into a finite number of sub-regions for each of which the condition (z, z0), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position of z0, within or upon the boundary of this sub-region. This is proved above as result (B).