Part 25

Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counter-clockwise; another square, also of side a, may be added to this, so that there is a side common; this common side being erased we have a composite region with a definite direction of perimeter; to this a third square of the same size may be attached, so that there is a side common to it and one of the former squares, and this common side may be erased. If this process be continued any number of times we obtain a region of the plane bounded by one or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of description, which is such that the region is on the left of the describing point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the greatest side of a triangle, and a lower limit for the smallest angle. In the former method, each square may be divided into four others by lines through its centre parallel to its sides; in the latter method each triangle may be divided into four others by lines joining the middle points of its sides; this halves the sides and preserves the angles. When we speak of a _region_ of the plane in general, unless the contrary is stated, we shall suppose it capable of being generated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of a _path_ in the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region.

There is a logical principle also which must be referred to. We frequently have cases where, about every interior or boundary, point z0 of a certain region a circle can be put, say of radius r0, such that for all points z of the region which are interior to this circle, for which, that is, |z - z0| < r0, a certain property holds. Assuming that to r0 is given the value which is the upper limit for z0, of the possible values, we may call the points |z - z0| < r0, the neighbourhood belonging to or _proper_ to z0, and may speak of the property as the property (z, z0). The value of r0 will in general vary with z0; what is in most cases of importance is the question whether the lower limit of r0 for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z, z0) is of a kind which we may call extensive; such, namely, that if it holds, for some position of z0 and all positions of z, within a certain region, then the property (z, z1) holds within a circle of radius R about any interior point z1 of this region for all points z for which the circle |z - z1| = R is within the region. Also in this case r0 varies continuously with z0. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) for _some_ point z0 within or upon the boundary of the sub-region, (2) for _every_ point z within or upon the boundary of the sub-region.

We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z, z0) holds for all points z and some point z0 of the region, be called _suitable_: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say [zeta]; after a certain stage all these will be interior to the proper region of [zeta]; this, however, is contrary to the supposition that they are all unsuitable.

We now make some applications of this result (B). Suppose a definite finite real value attached to every interior or boundary point of the region, say [f](x, y). It may have a finite upper limit H for the region, so that no point (x, y) exists for which [f](x, y) > H, but points (x, y) exist for which [f](x, y) > H - [epsilon], however small [epsilon] may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of [f](x, y) is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of [f](x, y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function [f](x, y) is the radius r0 of the neighbourhood proper to any point z0, spoken of above. We can hence prove the statement (A) above.

Suppose the property (z, z0) extensive, and, if possible, that the lower limit of r0 is zero. Let then [zeta] be a point such that the lower limit of r0 is zero for points z0 within a circle about [zeta] however small; let r be the radius of the neighbourhood proper to [zeta]; take z0 so that |z0 - [zeta]| < 1/2r; the property (z, z0), being extensive, holds within a circle, centre z0, of radius r - |z0 - [zeta]|, which is greater than |z0 - [zeta]|, and increases to r as |z0 - [zeta]| diminishes; this being true for all points z0 near [zeta], the lower limit of r0 is not zero for the neighbourhood of [zeta], contrary to what was supposed. This proves (A). Also, as is here shown that r0 [ = >] r - |z0-[zeta]|, may similarly be shown that r [=>] r0 - |z0 - [zeta]|. Thus r0 differs arbitrarily little from r when |z0-[zeta]| is sufficiently small; that is, r0 varies continuously with z0. Next suppose the function [f](x, y), which has a definite finite value at every point of the region considered, to be continuous but not necessarily real, so that about every point z0, within or upon the boundary of the region, [eta] being an arbitrary real positive quantity assigned beforehand, a circle is possible, so that for all points z of the region interior to this circle, we have |[f](x, y) - [f](x0, y0)| < 1/2[eta], and therefore (x', y') being any other point interior to this circle, |[f](x', y') - [f](x, y)| < [eta]. We can then apply the result (A) obtained above, taking for the neighbourhood proper to any point z0 the circular area within which, for any two points (x, y), (x', y'), we have |[f](x', y') - [f](x, y)| < [eta]. This is clearly an extensive property. Thus, a number r is assignable, greater than zero, such that, for any two points (x, y), (x', y') within a circle |z - z0| = r about any point z0, we have |[f](x', y') - [f](x, y)| < [eta], and, in particular, |[f](x, y) - [f](x0, y0)| < [eta], where [eta] is an arbitrary real positive quantity agreed upon beforehand.

Take now any path in the region, whose extreme points are z0, z, and let z1, ... z_(n - 1) be intermediate points of the path, in order; denote the continuous function [f](x, y) by [f](z), and let [f]_r denote any quantity such that |[f]_r - [f](z_r)| [=<] |[f](z_(r + 1)) - [f](z_r)|; consider the sum

(z1 - z0)[f]0 + (z2 - z1)[f]1 + ... + (z - z_(n - 1))[f](n - 1).

By the definition of a path we can suppose, n being large enough, that the intermediate points z1, ... z_(n - 1) are so taken that if z_i, z_(i + 1) be any two points intermediate, in order, to z_r and z_(r + 1), we have |z_(i + i) - z_i| < |z_(r + 1) - z_r|; we can thus suppose |z1 - z0|, |z2 - z1|, ... |z - z_(n - 1)|all to converge constantly to zero. This being so, we can show that the sum above has a definite limit. For this it is sufficient, as in the case of an integral of a function of one real variable, to prove this to be so when the convergence is obtained by taking new points of division intermediate to the former ones. If, however, z_(r, 1), z_(r, 2), ... z_(r, m - 1) be intermediate in order to z_r and z_(r + 1), and |[f]_(r, i) - [f](z_(r, i))| < |[f](z_(r, i + 1)) - [f](z_(r, i))|, the difference between [Sigma](z_(r + 1) - z_r)[f]_r and

[Sigma]{(z_(r, 1) - z_r)[f]_(r, 0) + (z_(r, 2) - z{r, 1})[f]_(r, 1) + ... + (z_(r + 1) - z_(r, m - 1))[f]_(r, m - 1)},

which is equal to

[Sigma]_r [Sigma]_i (z_(r, i + 1) - z_(r, i))([f]_(r, i) - [f]_r),

is, when |z_(r + 1) - z_r| is small enough, to ensure |[f](z_(r + 1)) - [f](z_r)| < [eta], less in absolute value than

[Sigma]2[eta] [Sigma] |z_(r, i + 1) - z{r, i}|,

which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 2[eta]S, and is therefore arbitrarily small.

The limit in question is called [int](z_0 to z) [f](z)dz. In

## particular when [f](z) = 1, it is obvious from the definition that its

value is z - z0; when [f](z) = z, by taking [f]_r = 1/2(z_(r + 1) - z_r), it is equally clear that its value is 1/2(z^2 - z0^2); 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 [f](z0), F(z0), such that, whatever real positive quantity [eta] may be, a real positive number [epsilon] exists for which the condition

| [f](z) - [f](z0) | | ---------------- - F(z0) | < [eta], | 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| < [epsilon]. Then [f](z0) is called a differentiable function of the complex variable z0 over this region, its differential coefficient being F(z0). The function [f](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 [eta] to be retained the same for all points z0 of the region, and [sigma]0 to be the upper limit of the possible values of [epsilon] for the point z0, it is to be presumed that [sigma]0 will vary with z0, and it is not obvious as yet that the lower limit of the values of [sigma]0 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).

Hence it can be proved that, for a differentiable function [f](z), the integral [int](z_1 to z) [f](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 [int][f](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

[f](z) = [f](z0) + (z - z0)F(z0) + [eta][theta](z - z0), where |[theta]| < 1,

we have _ _ _ _ / / / / |[f](z)dz = [[f](z0) - z0F(z0)] |dz + F(z0) |zdz + [eta] |[theta](z - z0)dz, _/ _/ _/ _/

which, as the path is closed, is [eta] [int][theta](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 [eta]ap, where a is the greatest side of the triangle and p is its perimeter; if [Delta] be the area of the triangle, we have [Delta] = 1/2ab sin C > ([alpha]/[pi])ba, where [alpha] is the least angle of the triangle, and hence a(a + b + c) < 2a(b + c) < 4[pi][Delta]/[alpha]; the integral [int][f](z)dz round the perimeter of the triangle is thus < 4[pi][eta][Delta]/[alpha]. 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 [int][f](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 4[pi][eta]K/[alpha], where K is the whole area of the region, and [alpha] is the smallest angle of the component triangles. However small [eta] 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 [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.

S 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.

(3) Hence if the series u0(z) + u1(z) + ... to [oo] 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 / 2[pi]i) [int] (2[pi]i / (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 [f](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, _ _ 1 / [f](t)dt 1 / [f](t)dt [f](z0) = ------ | -------- - ------ | --------, 2[pi]i _/R t - z0 2[pi]i _/r t - z0

where the integrals are both counter-clockwise round the two circumferences respectively; putting in the first (t - z0)^(-1) = [Sigma]_(n=0) z0^n/t^(n + 1), and in the second (t - z0)^(-1) = [Sigma]_(n=0) t^n/z0^(n + 1), we find [f](z0) = [Sigma] (-[oo] to [oo]) A_nz0^n, wherein A_n = (1 / 2[pi]i) [int] [f(t) / t^(n + 1)] dt, taken round any circle, centre the origin, of radius intermediate between r and R. Particular cases are: ([alpha]) when the region of definition of the function includes the whole interior of the outer circle; then we may take r = 0, the coefficients A_n for which n < 0 all vanish, and the function [f](z0) is expressed for the whole interior |z0| < R by a power series [Sigma] (0 to [oo]) A_n z0^n. 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.

([beta]) 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 z^m [f](z) has a finite limit when |z| diminishes to zero, all the coefficients A_n for which n < -m vanish, and we have

f(z0) = A_(-m) z0^(-m) + A_(-m + 1) z0^(-m + 1) + ... + A_(-1) z0^(-1) + A0 + A1z0 ... to [oo].

Such a case occurs, for instance, when [f](z) = cosec z, the number m being unity.

S 6. _Singular Points._--The _region of existence_ of a differentiable function of z is an unclosed aggregate of points, each of which is an interior point of a neighbourhood consisting wholly of points of the aggregate, at every point of which the function is definite and finite and possesses a unique finite differential coefficient. Every point of the plane, not belonging to the aggregate, which is a limiting point of points of the aggregate, such, that is, that points of the aggregate lie in every neighbourhood of this, is called a _singular point_ of the function.