Page:EB1911 - Volume 11.djvu/328

sinπx/πx, we have

sin πx	= $\prod _{\infty }^{-\infty }$ { (1 −	x	) exp (	x	) },
πx		n		n

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

1	= xeCx $\prod _{n=1}^{\infty }$ { (1 +	x	) exp ( −	x	) },
Γ(x)		n		n

where C is a constant, and Γ(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) = Σa_nzⁿ 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 a_i there be associated a polynomial in (z − a_i)⁻¹, say g_i; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point a_i a pole whereat the expansion consists of the terms g_i, together with a power series in z − a_i; the function is expressible as an infinite series of terms g_i − γ_i, where γ_i is also a rational function.

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

$\prod _{n=1}^{\infty }$ ( 1 −	an − cn	) r n exp (gn),
	z − cn

where with every point a_n is associated a proper point c_n of (c), and

gn = rn Σ μ ns=1	1	(	an − cn	) s,
	s		z − cn

μ_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 a_n = R(1 − n⁻¹) exp (iπ √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

|	a − b	| < ε, for |a − b| < η;
	z − b

the rational function of the complex variable t,

1	[ 1 − (	a − b	)n ],
t − a		t − a

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

F = Σ Ap {	1	[ 1 − (	a − b	) n ] } p,
	t − a		t − b

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

ƒ = Σ A_p(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

ƒ(z) =	1	∫	ƒ(t)dt	,
	2πi		t − z

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

S =	1	Σ	ƒ(ti ) (ti+1 − ti )	,
	2πi		ti  − z

where the points t_i 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 ti+1 − t_i, has been carried sufficiently far, that

|S − ƒ(z)| < ε,

for all points z of R₀, where ε 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 Γ 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 Γ. 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 ε₁, ε₂, ... to be an indefinitely continued sequence of real positive numbers, converging to zero, and P_r to be the polynomial such that, within C₀, |P_r − ƒ(z)| < ε_r; then the infinite series of polynomials

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

whose sum to n terms is P_n(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 Γ 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 Γ. 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 Γ.

§ 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 ζ = ξ + iη, enclose the points ζ = l, ζ = 1 + c by means of (i.) the straight lines η = ±a, from ξ = l to ξ = 1 + c, (ii.) a semicircle convex to ζ = 0 of equation (ξ − 1)² + η² = a², (iii.) a semicircle concave to ζ = 0 of equation (ξ − 1 − c)² + η² = 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 σ = 1/r(c/a). Now take

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