Part 26
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/[zeta], 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 [zeta]; in such case we say that the region of existence of the function contains the point z = [oo]. A series in z^(-1) has a finite limit when |z| = [oo]; 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 [Sigma]a_n z^n 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 = [oo]; such is, for instance, the case of the function exp (z) = [Sigma]z^n/n!. This may be regarded as a particular case of a well-known result (S 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 [Sigma]_(n=0) z^m, where m = n^2, the series [Sigma]_(n=0) z^m where m = n!, and the series [Sigma](n=1 to 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 been proved that the value of the function at all points of its region of existence can be obtained from its value, supposed given by a series in one original circle, by a succession of such processes of analytical continuation.
S 7. _Monogenic Functions._--This suggests an entirely different way of formulating the fundamental parts of the theory of functions of a complex variable, which appears to be preferable to that so far followed here.
Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z - z0, about any point z0 interior to its circle of convergence, and the new series converges certainly for |z - z0| < r - |z0|, if r be the original radius of convergence. If for every position of z0 this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence. If for some position of z0 the derived series converges for |z - z0| < r - |z0| + D, then it can be shown that for points z, interior to the original circle, lying in the annulus r - |z0| < |z - z0| < r - |z0| + D, the value represented by the derived series agrees with that represented by the original series. If for another point z1 interior to the original circle the derived series converges for |z - z1| < r - |z1| + E, and the two circles |z - z0| = r - |z0| + D, |z - z1| = r - |z1| + E have interior points common, lying beyond |z| = r, then it can be shown that the values represented by these series at these common points agree. Either series then can be used to furnish an analytical continuation of the function as originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined by an aggregate of power series of which every one has points of convergence common with some one or more others; the whole aggregate of points of the plane which can be so reached constitutes the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have radii of convergence diminishing indefinitely to zero; these are the singular points. The circle of convergence of any of the series has at least one such singular point upon its circumference. So regarded the function is called a _monogenic_ function, the epithet having reference to the single origin, by one power series, of the expressions representing the function; it is also sometimes called a _monogenic analytical_ function, or simply an _analytical_ function; all that is necessary to define it is the value of the function and of all its differential coefficients, at some one point of the plane; in the method previously followed here it was necessary to suppose the function differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, that [int][f](z)dz = 0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued--it may quite well happen that starting from a
## particular power series, converging over a certain circle, and
applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The notion of basing the theory of functions on the theory of power series is, after Newton, largely due to Lagrange, who has some interesting remarks in this regard at the beginning of his _Theorie des fonctions analytiques_. He applies the idea, however, primarily to functions of a real variable for which the expression by power series is only of very limited validity; for functions of a complex variable probably the systematization of the theory owes most to Weierstrass, whose use of the word monogenic is that adopted above. In what follows we generally suppose this point of view to be regarded as fundamental.
S 8. _Some Elementary Properties of Single Valued Functions._--A _pole_ is a singular point of the function [f](z) which is not a singularity of the function 1/[f](z); this latter function is therefore, by the definition, capable of representation about this point, z0, by a series [[f](z)]^(-1 ) = [Sigma]a_n (z - z0)^n. If herein a0 is not zero we can hence derive a representation for [f](z) as a power series about z0, contrary to the hypothesis that z0 is a singular point for this function. Hence a0 = 0; suppose also a1 = 0, a2 = 0, ... a_(m - 1) = 0, but a_m [+-] 0. Then [[f](z)]^(-1) = (z - z0)^m [a_m + a_(m + 1)(z - z0) + ...], and hence (z - z0)^m [f](z) = a_m^(-1) + [Sigma]b_n (z - z0)^n, namely, the expression of [f](z) about z = z0 contains a finite number of negative powers of z - z0 and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.
The integral [int][f](z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 2[pi]iA1, where A1 is the coefficient of (z - z0)^(-1) in the expansion of [f](z) at the pole; this coefficient has therefore a certain uniqueness, and it is called the _residue of [f](z) at the pole_. Considering a region in which there are no other singularities than poles, all these being interior points, _the integral (1 / 2[pi]i) [int][f](z)dz round the boundary of this region is equal to the sum of the residues at the included poles_, a very important result. Any singular point of a function which is not a pole is called an _essential singularity_; if it be isolated the function is capable, in the neighbourhood of this point, of approaching arbitrarily near to any assigned value. For, the point being isolated, the function can be represented, in its neighbourhood, as we have proved, by a series [Sigma] (-[oo] to [oo]) a_n(z - z0)^n; it thus cannot remain finite in the immediate neighbourhood of the point. The point is necessarily an isolated essential singularity also of the function {[f](z) - A}^(-1) for if this were expressible by a power series about the point, so would also the function [f](z) be; as {[f](z) - A}^(-1) approaches infinity, so does [f](z) approach the arbitrary value A. Similar remarks apply to the point z = [oo], the function being regarded as a function of [zeta] = z^(-1). In the neighbourhood of an essential singularity, which is a limiting point also of poles, the function clearly becomes infinite. For an essential singularity which is not isolated the same result does not necessarily hold.
A single valued function is said to be an _integral_ function when it has no singular points except z = [oo]. Such is, for instance, an integral polynomial, which has z = [oo] for a pole, and the functions exp (z) which has z = [oo] as an essential singularity. A function which has no singular points for finite values of z other than poles is called a _meromorphic_ function. If it also have a pole at z = [oo] it is a _rational_ function; for then, if a1, ... a_s be its finite poles, of orders m1; m2, ... m_s, the product (z - a1)^m1 ... (z - a_s)^m_s[f](z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (z^ - 1)^(-m) [Sigma]_(r=0) a_r(z^ - 1)^r; thus (z - a1)^m1 ... (z - a_s)^m_s[f](z) is capable of a form [Sigma]_(r=0) b_r z^r, but z^(-m) [Sigma]_(r=0) b_r z^r remains finite for z = [oo]. Therefore b_(r + 1) = b_(r + 2) = ... = 0, and[f](z) is a rational function.
If for a single valued function F(z) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to be a1, a2, a3, ... with |a1| [=<] |a2| [=<] |a3| ... and limit |a_n| = [oo]. About a_s the function is expressible as [Sigma] (-[oo] to [oo]) A_n(z - a_s)^n; let [f]_s(z) = [Sigma] (-[oo] to 1) A^n(z - a_s)^n be the sum of the negative powers in this expansion. Assuming z = 0 not to be a singular point, let [f]_s(z) be expanded in powers of z, in the form [Sigma]_(n=0) C_n z^n, and [mu]_s be chosen so that F_s(z) = [f]_s(z) - [Sigma] (1 to [mu]_s-1) C_nz^n = [Sigma] ([mu]_s to [oo]) C_n z^n is, for |z| < r_s < |a_s|, less in absolute value than the general term [epsilon]_s of a fore-agreed convergent series of real positive terms. Then the series [phi](z) = [Sigma] (s=1 to [oo]) F_s(z) converges uniformly in any finite region of the plane, other than at the points a_s, and is expressible about any point by a power series, and near a_s, [phi](z) - f_s(z) is expressible by a power series in z-a_s. Thus F(z) - [phi](z) is an integral function. In
## particular when all the finite singularities of F(z) are poles, F(z)
is hereby expressed as the sum of an integral function and a series of rational functions. The condition |F_s(z)| < [epsilon]_s is imposed only to render the series [Sigma]F_s(z) uniformly convergent; this condition may in particular cases be satisfied by a series [Sigma] G_s(z) where G_s(z) = [f]_s(z) - [Sigma] (1 to [nu]_s-1) C_nz^n and [nu]_s < [mu]_s. An example of the theorem is the function [pi] cot [pi]z - z^(-1) for which, taking at first only half the poles, [f]_s(z) = 1/(z-s); in this case the series [Sigma]F_s(z) where F_s(z) = (z - s)^-1 + s^-1 is uniformly convergent; thus [pi]cot[pi]z - z^-1 - [Sigma] (-[oo] to [oo]) [(z - s)^-1 + s^-1], where s = 0 is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.
Considering an integral function [f](z), if there be no finite positions of z for which this function vanishes, the function [lambda][[f](z)] is at once seen to be an integral function, [phi](z), or [f](z) = exp[[phi](z)]; if however great R may be there be only a finite number of values of z for which [f](z) vanishes, say z = a1, ... a_m, then it is at once seen that [f](z) = exp [[phi](z)].(z - a1)^h1...(z - a_m)^h_m, where [phi](z) is an integral function, and h1, ... h_m are positive integers. If, however, [f](z) vanish for z = a1, a2 ... where |a1| [=<] |a2| [=<] ... and limit |a_n| = [oo], and if for simplicity we assume that z - 0 is not a zero and all the zeros a1, a2, ... are of the first order, we find, by applying the preceding theorem to the function (1 / [f](z)) (d[f](z) / dz), that [f](z) = exp[[phi](z)] [Pi] (n=1 to [oo]) {(1 - z/a_n) exp[phi]_n(z)}, where [phi](z) is an integral function, and [phi]_n(z) is an integral polynomial of the form
z z^2 z^s [phi]_n(z) = --- + ------ + ... + ------. a_n 2a^2_n sa_n^s
The number s may be the same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to take s = n. In
## particular for the function sin[pi]x / [pi]x, we have
_ _ sin[pi]x [oo] | / x \ / x \ | -------- = [Pi] | ( 1 - -- ) exp ( -- ) |, [pi]x -[oo] |_ \ n / \ n / _|
where n = 0 is excluded from the product. Or again we have _ _ 1 [oo] | / x \ / x \ | ---------- = xe^C_x [Pi] | ( 1 - -- ) exp ( - -- ) |, [Gamma](x) n=1 |_ \ n / \ n / _|
where C is a constant, and [Gamma](x) is a function expressible when x is real and positive by the integral [int] (0 to [oo]) 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 [f](z), and the law of increase of the coefficients in the series [f](z) = [Sigma] a_n z^n 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 S 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 = [oo] 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 - [gamma]_i, where [gamma]_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
[oo] / a_n - c_n \^r_n [Pi] ( 1 - --------- ) exp(g_n), n=1 \ z - c_n /
where with every point a_n is associated a proper point c_n of (c), and
_[mu]_n \ 1 / a_n - c_n \^s g_n = r_n /_ -- ( --------- ), s=1 s \ z - c_n /
[mu]_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(i[pi] [root]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.
S 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.
S 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 | | ----- | < [epsilon], for |a - b| < [eta]; | z - b |
the rational function of the complex variable t, _ _ 1 | / a - b \^n | ----- |1 - ( ----- ) |, t - a |_ \ t - b / _|
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
_ _ _ \ { 1 | / a - b \^n | }^p F = /_ A_p { ----- |1 - ( ----- ) | }, { 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 _ \ [f] = /_ A_p (t - a)^(-p),
and differing, outside R or upon the boundary of R, from [f], in the fact that while [f] 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 [f](t) whose region of definition includes C and the interior of R can be represented at all points z in R[0] by _ 1 / [f](t)dt [f](z) = ------ | --------, 2[pi]i _/ t - z
where the path of integration is C. This integral is the limit of a sum _ 1 \ [f](t_i) (t_(i + 1) - t_i) S = ------ /_ --------------------------, 2[pi]i t_i - 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 [f](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) - t_i, has been carried sufficiently far, that
| S - [f](z) | < [epsilon],
for all points z of R0, where [epsilon] 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 [f](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 [GAMMA] 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 [f](z) for all points of R0 whose poles are at one finite point c external to [GAMMA]. 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 [epsilon]1, [epsilon]2, ... 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 - [f](z)| < [epsilon]_r; then the infinite series of polynomials
P1(z) + {P2(z) - P1(z)} + {P3(z) - P2(z)} + ...,