where n = 0 is excluded from the product. Or again we have
| 1 | = xeCx Π ∞n=1 { (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 ∫ ∞0 e−t tx−1dt.
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) = Σ 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 there be associated a polynomial in (z − ai)−1, say gi; 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 a pole whereat the expansion consists of the terms gi, together with a power series in z − ai; the function is expressible as an infinite series of terms gi − γi, where γi is also a rational function.
II. With a similar aggregate (a), with limiting points (c), suppose with every point ai there is associated a positive integer ri. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order ri at the point ai, but not elsewhere, expressible in the form
| Π ∞n=1 ( 1 − | an − cn | ) r n exp (gn), |
| z − cn |
where with every point an is associated a proper point cn of (c), and
| gn = rn Σ μ ns=1 | 1 | ( | an − cn | ) s, |
| s | z − cn |