Page:EB1911 - Volume 11.djvu/328

Rh $sinx⁄x$, we have

where n = 0 is excluded from the product. Or again we have

where C is a constant, and (x) is a function expressible when x is real and positive by the integral $$\int_0^\infty e^{-t}t^{x-1}dt$$.

There exist interesting investigations as to the connexion of the value of s above, the law of increase of the modulus of the integral function ƒ(z), and the law of increase of the coefficients in the series ƒ(z) = anzn as n increases (see the bibliography below under Integral Functions). It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional cases, one value at most. For instance, the function exp (z) assumes every finite value except zero (see below under § 21, Modular Functions).

The two theorems given above, the one, known as Mittag-Leffler’s theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point z = ∞ as their only limiting point, the other, Weierstrass’s factor theorem, giving the expression of an integral function as a product of factors each with only one zero in the finite part of the plane, may be respectively generalized as follows:—

I. If a1, a2, a3, ... be an infinite series of isolated points having the points of the aggregate (c) as their limiting points, so that in any neighbourhood of a point of (c) there exists an infinite number of the points a1, a2, ..., and with every point ai&#8202; there be associated a polynomial in (z − ai&#8202;)−1, say gi&#8202;; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point ai&#8202; a pole whereat the expansion consists of the terms gi&#8202;, together with a power series in z − ai&#8202;; the function is expressible as an infinite series of terms gi&#8202; − i&#8202;, where i&#8202; is also a rational function.

II. With a similar aggregate (a), with limiting points (c), suppose with every point ai&#8202; there is associated a positive integer ri&#8202;. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order ri&#8202; at the point ai&#8202;, but not elsewhere, expressible in the form

where with every point an is associated a proper point cn of (c), and

n being a properly chosen positive integer.

If it should happen that the points (c) determine a path dividing the plane into separated regions, as, for instance, if an = R(1 − n−1) exp (i √2.n), when (c) consists of the points of the circle |z| = R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.

§ 9. Construction of a Monogenic Function with a given Region of Existence.—A series of isolated points interior to a given region can be constructed in infinitely many ways whose limiting points are the boundary points of the region, or are boundary points of the region of such denseness that one of them is found in the neighbourhood of every point of the boundary, however small. Then the application of the last enunciated theorem gives rise to a function having no singularities in the interior of the region, but having a singularity in a boundary point in every small neighbourhood of every boundary point; this function has the given region as region of existence.

§ 10. Expression of a Monogenic Function by means of Rational Functions in a given Region.—Suppose that we have a region R0 of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called C0. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of C0; let a, b, ... be finite points upon C or outside R. Then when b is near enough to a, the fraction (a − b)/(z − b) is arbitrarily small for all positions of z; say

the rational function of the complex variable t,

in which n is a positive integer, is not infinite at t = a, but has a pole at t = b. By taking n large enough, the value of this function, for all positions z of t belonging to R0, differs as little as may be desired from (t − a)−1. By taking a sum of terms such as

we can thus build a rational function differing, in value, in R0, as little as may be desired from a given rational function

ƒ = Ap(t − a)−p,

and differing, outside R or upon the boundary of R, from ƒ, in the fact that while ƒ is infinite at t = a, F is infinite only at t = b. By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0 arbitrarily little from the former, whose poles are all at the point c.

Now any monogenic function ƒ(t) whose region of definition includes C and the interior of R can be represented at all points z in R0 by

where the path of integration is C. This integral is the limit of a sum

where the points ti&#8202; are upon C; and the proof we have given of the existence of the limit shows that the sum S converges to ƒ(z) uniformly in regard to z, when z is in R0, so that we can suppose, when the subdivision of C into intervals t i+1 − ti&#8202;, has been carried sufficiently far, that

|S − ƒ(z)| < ,

for all points z of R0, where is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to R0. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from ƒ(z), for all points z of R0, with poles at arbitrary positions outside R0 which can be reached by finite continuous curves lying outside R from the points of C.

In particular, to take the simplest case, if C0, C be simple closed polygons, and be a path to which C approximates by taking the number of sides of C continually greater, we can find a rational function differing arbitrarily little from ƒ(z) for all points of R0 whose poles are at one finite point c external to. By a transformation of the form t − c = r−1, with the appropriate change in the rational function, we can suppose this point c to be at infinity, in which case the rational function becomes a polynomial. Suppose 1, 2, ... to be an indefinitely continued sequence of real positive numbers, converging to zero, and Pr to be the polynomial such that, within C0, |Pr − ƒ(z)| < r; then the infinite series of polynomials

P1(z) + {P2(z) − P1(z)} + {P3(z) − P2(z)} + ...,

whose sum to n terms is Pn(z), converges for all finite values of z and represents ƒ(z) within C0.

When C consists of a series of disconnected polygons, some of which may include others, and, by increasing indefinitely the number of sides of the polygons C, the points C become the boundary points of a region, we can suppose the poles of the rational function, constructed to approximate to ƒ(z) within R0, to be at points of. A series of rational functions of the form

H1(z) + {H2(z) − H1(z)} + {H3(z) − H2(z)} + ...

then, as before, represents ƒ(z) within R0. And R0 may be taken to coincide as nearly as desired with the interior of the region bounded by.

§ 11. Expression of (1 − z)−1 ''by means of Polynomials. Applications.''—We pursue the ideas just cursorily explained in some further detail.

Let c be an arbitrary real positive quantity; putting the complex variable =  + i, enclose the points  = l,  = 1 + c by means of (i.) the straight lines = ±a, from  = l to  = 1 + c, (ii.) a semicircle convex to = 0 of equation ( − 1)2 + 2 = a2, (iii.) a semicircle concave to = 0 of equation ( − 1 − c)2 + 2 = a2. The quantities c and a are to remain fixed. Take a positive integer r so that $1⁄r$($c⁄a$) is less than unity, and put = $1⁄r$($c⁄a$). Now take

c1 = 1 + c/r, c2 = 1 + 2c/r, ... cr = 1 + c;