Page:EB1911 - Volume 11.djvu/326

variables x₀, y₀, where $z_{0}=x_{0}+iy_{0}$, 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).

Hence it can be proved that, for a differentiable function $f(z)$, the integral $\int _{z_{1}}^{z}f(z)dz$ 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

$f(z)=f(z_{0})+(z-z_{0})F(z_{0})+\eta \theta (z-z_{0})$, where $|\theta |<1$,

we have

$$\int f(z)dz=[f(z_{0})-z_{0}F(z_{0})]\int dz+F(z_{0})\int zdz+\eta \int \theta (z-z_{0})dz,$$

which, as the path is closed, is $\eta \int \theta (z-z_{0})dz$. 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 Δ be the area of the triangle, we have $\Delta ={\frac {1}{2}}ab\sin C>(\alpha /\pi )ba$, where α 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 $<4\pi \Delta /\alpha$. 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 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 $f(z)$ 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 $f(z)/(z-z_{0})$, 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 $\int {\frac {dzf(z)}{z-z_{0}}}$ 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 $z-z_{0}=rE(i\theta )$, $dz/(z-z_{0})=id\theta$, and $f(z)$ differs arbitrarily little from $f(z_{0})$ if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to $2\pi if(z_{0})$. Hence $f(z_{0})={\frac {1}{2\pi i}}\int {\frac {dtf(t)}{t-z_{0}}}$, where this integral is round the boundary of the original region. From this it appears that

$$F(z_{0})=\lim {\frac {f(z)-f(z_{0})}{z-z_{0}}}={\frac {1}{2\pi i}}\int {\frac {dtf(t)}{(t-z_{0})^{2}}}$$

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 $S(z)=u_{0}(z)+u_{1}(z)+\dots +u_{n-1}(z)+R_{n}(z)$, where $|R_{n}(z)|<\epsilon$ for all points of the path, we have

$$\int _{z_{0}}^{z}S(z)dz=\int _{z_{0}}^{z}u_{0}(z)dz+\int _{z_{0}}^{z}u_{1}(z)dz+\dots +\int _{z_{0}}^{z}u_{n-1}(z)dz+\int _{z_{0}}^{z}R_{n}(z)dz,$$

wherein, in absolute value, $\int _{z_{0}}^{z}R_{n}(z)dz<\epsilon L$, 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 $\psi (z)=\int _{z_{0}}^{z}f(x,y)dz$, 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 ∞ 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 ∞−∞ A_nz₀ⁿ, wherein A_n = (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 A_n 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 A_nz₀ⁿ. 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 z^mƒ(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−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.

§ 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.

About every interior point z₀ of the region of existence the function may be represented by a power series in z − z₀, and the series converges and represents the function over any circle centre at z₀ which contains no singular point in its interior. This has been proved above. And it can be similarly proved, putting z = 1/ζ, that if the region of existence of the function contains all points of the plane for which |z| > R, then the function is representable for all such points by a power series in z− 1 or ζ; in such case we say that the region of existence of the function contains the point z = ∞. A series in z− 1 has a finite limit when |z| = ∞; a series in z cannot remain finite for all points z for which |z| > R; for if, for |z| = R, the sum of a power series Σa_nzⁿ in z is in absolute value less than M, we have |a_n| < Mr⁻ⁿ, and therefore, if M remains finite for all values of r however great, a_n = 0. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = ∞; such is, for instance, the case of the function exp (z) = Σzⁿ/n!. This may be regarded as a particular case of a well-known result (§ 7), that the circumference of convergence of any power series representing the function contains at least one singular point. As an extreme case functions exist whose region of existence is circular, there being a singular point in every arc of the circumference, however small; for instance, this is the case for the functions represented for |z| < 1 by the series $\sum _{n=0}z^{m}$, where m = n², the series $\sum _{n=0}z^{m}$ where m = n!, and the series $\sum _{n=1}^{n=0}z^{m}/(m+1)(m+2)$ where $m=a^{n}$, $a$ being a positive integer, although in the last case the series actually converges for every point of the circle of convergence |z| = 1. If z be a point interior to the circle of convergence of a series representing the function, the series may be rearranged in powers of z − z₀; as z₀ approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in z − z₀ diminish to zero; when, however, a circle can be put about z₀, not containing any singular point of the function, but containing points outside the circle of convergence of the original series, then the series in z − z₀ gives the value of the function for these external points. If the function be supposed to be given only for the interior of the original circle, by the original power series, the series in z − z₀ converging beyond the original circle gives what is known as an analytical continuation of the function. It appears from what has