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variables to, yo, where zoxo+iy, over the region; it will appear that F (2) is also continuous and in fact also a differentiable function

of zo.

Supposing to be retained the same for all points zo of the region, and to be the upper limit of the possible values of e for the point zo, it is to be presumed that will vary with so, and it is not obvious as yet that the lower limit of the values of o as z 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 (2, 2), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position of zo, 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 f(2)dz has the same value by whatever path within the region we pass from z to z. This we prove by showing that when taken round a closed path in the region the integral ff()dz vanishes, Consider first a triangle over which the condition (2, 2) holds, for some position of 2 and every position of s, within or upon the boundary of the triangle. Then as

we have

f(z) = f(20)+(z−20)F(z)+n9(z−zo), where[0]<1, ff(z)dz=[f(20)—20F (30)][ds+F(20){zdz+nƒ0 (z−20)dz, which, as the path is closed, is fe(3-2)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 nap, where a is the greatest side of the triangle and p is its perimeter; if A be the area of the triangle, we have lab sin C>(a/r)ba, where a is the least angle of the triangle, and hence a(a+b+c)<2a(b+c) <4A/a; the integral ff(z)ds round the perimeter of the triangle is thus <4/a. Now consider any region made up of triangles, as before explained, in each of which the condition (, z) holds, as in the triangle just taken. The integral ff()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 4K/a, where K is the whole area of the region, and a is the smallest angle of the component triangles. However small 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(2) at the boundary points. For consider in the original region the function f(z)/(2-20), where zo is an interior point: this satisfies the same conditions as f(z) except in the immediate neighbourhood of 50. Taking out then from the original region a small regular polygonal region with zo as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral (2) round the boundary of the original region is equal to the same integral taken counterclockwise round a small circle having zo as centre: on this circle, however, if -20=7E(10), dz/(z-zo)=ide, and f(2) differs arbitrarily little from f(z) if is sufficiently small; the value of the integral round this circle is therefore, ultimately, when vanishes, equal to "dtf(t) 2=if(50). Hence f(2) = where this integral is round the 1-20' boundary of the original region. From this it

F(20) lim.

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that
appears
ƒ(2) —ƒ(20)=;
dif(t)
2-30 2T1) (-20)

also round the boundary of the original region. This form shows, however, that F(e) is a continuous, finite, differentiable function of ze over the whole interior of the original region.

§ 5. Applications. The previous results have manifold appli

cations.

(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(2) = 20 (2) + u1(z) +...+ -1(2)+R(2), where | R.(z) | < for all points of the path, we have SS(e)ds=fuo (z)dz+f'u, (e) dz++fumi (2)dz + "Ru(t)ds, wherein, in absolute value, R. (2)dz <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 regionff(x, y)dz=o, then

|

() = f(x, y)ds, for interior points zo, z, is a differentiable function of 2, having for its differential coefficient the function f (x, y), which is therefore also a differentiable function of s at interior points. (3) Hence if the series to(s) + (2) +... to be uniformly convergent over a region, its terms being differentiable functions of 2, then its sum S() is a differentiable function of z, whose differential coefficient, given by, is obtainable by differentiating the series. This theorem, unlike (1), does not hold for functions of a 2x1) (1-2)2' real variable.

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(4) If the region of definition of a differentiable function f(z) include the region bounded by two concentric circles of radii r, R, with centre at the origin, and be an interior point of this region, where the integrals are both counter. clockwise round the two circumferences respectively; putting in the first (-20)-1= Σ 20"/¿111, and in the second (-20)—1 = — Σ t^/20+1,

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31

21

n-0

we find ƒ(20) := £_ Anzo”, wherein A.=d, taken round any circle, centre the origin, of radius intermediate between ▾ and R. Particular cases are: (a) when the region of definition of the function includes the whole interior of the outer circle; then we may take o, the coefficients A, for which <o all vanish, and the function f(20) is expressed for the whole interior |zo|<R by a power scries Σ Anzo". In other words, about every interior point c of the region of definition a differentiable function of 2 is expressible by a power series in 2-c; a very important result.

extends to within arbitrary nearness of this on all sides, and at the (8) If the region of definition, though not including the origin, same time the product (2) has a finite limit when a diminishes to zero, all the coefficients An for which n<-m vanish, and we have ƒ(50) = A_m20¬TM+A+++...+A_120 ̃1+Ao +Az...to ∞. Such a case occurs, for instance, when f(z) = cosec z, the number m being unity.

一個

function of z is an unclosed aggregate of points, each of which § 6. Singular Points.-The region of existence of a differentiable 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 zo of the region of existence the function may be represented by a power series in z-20, and the series converges and represents the function over any circle centre at zo which contains no singular point in its interior. This has been that if the region of existence of the function contains all points of proved above. And it can be similarly proved, putting = 1/5, the plane for which [>R, then the function is representable for all such points by a power series in or ; in such case we say that the region of existence of the function contains the point z = ∞. A series in has a finite limit when [2]; a series in z cannot the sum of a power series 2a in z is in absolute value less than M, remain finite for all points 2 for which >R; for if, for [z]=R, we have la.<Mr", and therefore, if M remains finite for all values of however great, d=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) = Ez3/n!. This may be regarded as a particular case of a well-known result (87), 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 < 1 by the series Σ_ z", where m=n2, the series E 2-0

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where m=n!, and the series / (+1)(m+2) where m=a",

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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-20; as 20 approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in

so diminish to zero; when, however, a circle can be put about 20, 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-zo 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-20 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.

§ 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.

hold.

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
a.(z-20)"; it thus cannot remain finite in the immediate neighbour.
hood of the point. The point is necessarily an isolated essential
singularity also of the function f()-A), for if this were expressible
by a power series about the point, so would also the function f()
be; as {f(2)-A) approaches infinity, so does f(2) approach the
arbitrary value A. Similar remarks apply to the point =, the
hood of an essential singularity, which is a limiting point also of
function being regarded as a function of . In the neighbour-
poles, the function clearly becomes infinite. For an essential singu-
larity which is not isolated the same result does not necessarily
A single valued function is said to be an integral function
when it has no singular points except z=∞. Such is, for
instance, an integral polynomial, which has ∞ for a pole, and
the functions exp (2) which has 2 as an essential singularity.
A function which has no singular points for finite values of
other than poles is called a meromorphic function. If it also
have a pole at = it is a rational function; for then, if
1, ... a, be its finite poles, of orders my, my,...
m., the
product (2-a1)TM1. . . (z—a.)TM‚f(z) is an integral function with
a pole at infinity, capable therefore, for large values of z, of an
expression (2-1)- Σ 4,(2 ̄1)"; thus (2-c1), . . . (z—a,)TM √(2)
is capable of a form Σ b, but - b remains finite for
=∞. Therefore b,+1=b,+2= ... =0, and f(2) is a rational
function.

Starting with a convergent power series, say in powers of s, this series can be arranged in powers of z-zo, about any point zo interior to its circle of convergence, and the new series converges certainly for 12-2017-120), if be the original radius of convergence. If for every position of 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 zo the derived series converges for 1-20 <7-120+D, then it can be shown that for points z, interior to the original circle, lying in the annulus <12-20|<- |20|+D, the value represented by the derived series agrees with that represented by the original series. If for another point z interior to the original circle the derived series converges for 3-<r-+E, and the two circles 13-20 7-12+D, 12-21|=r-|21|+E have interior points common, lying beyond 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 functions 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 functional. differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, that ff(c)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 Théorie 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.

§ 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(2); this latter function is therefore, by the definition, capable of representation about this point, zo, by a series [ƒ(2)] ̄1=2a,(2—20)". If herein ao is not zero we can hence derive a representation for f(2) as a power series about zo, contrary to the hypothesis that zo is a singular point for this function. Hence 400; suppose also a1 =0, a2 =0,... am-1=0, but am±0. Then [f(2)=(2—20)" (an+am+1 (2—20)+...], and hence (3—30)TM√(2) = a+2b.(-zo)", namely, the expression of f(z) about z=20 contains a finite number of negative powers of z-zo and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.

The integral ff(z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 21A, where A, is the coefficient of (2-2) 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 (f(2)ds round the boundary of this region is equal to

let f.(2)

=

1

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0

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.

and limit

A(z-a,)";

If for a single valued function F(2) 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 a, a2, as,... with lalalal About 4. the function is expressible as A"(-a.)" be the sum of the negative powers in this expansion. Assuming =0 not to be a singular point, let f,(z) be expanded in powers of z, in the form E C.2", and, be chosen so that F,(3)=f.(z) — “E C„z" = £ C22′′is, for |2| <7.<\a.), less in absolute value than the general term e, of a fore-agreed convergent series of real positive terms. Then the series () = F.(z) converges uniformly in any finite region of the plane, other than at the points a,, and is expressible about any point by a power series, and near a., (c)-f.(2) is expressible by a power series in 2-a Thus ()-(2) is an integral function. In particular when all the finite singularities of F(2) are poles, F(s) is hereby expressed as the sum of an integral function and a series of rational functions. The condition (F.(2) < is imposed only to render the series F. (2) uniformly convergent; this condition may in particular cases be satisfied by a series ZG,(2) where G.(z) =f.() - C" and v.<μa. An example of the theorem is the function cot #z-z1 for which, taking at first only half the poles, f(z) = 1/(2-5); in this case the series ZF,(z) where F.(2)(2-5)+s is uniformly convergent: thus cot #2-21-2 ((2-5)+5'], where s=0 is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.

of

81

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1

Considering an integral function f(2), if there be no finite positions for which this function vanishes, the function [f(z)] is at once seen to be an integral function, (2), or ƒ(z) = exp [(2)]; if however great R may be there be only a finite number of values of z for which (2) vanishes, say z=a,...am, then it is at once scen that f(z) = exp ()]. (-a)... (-a)m, where (2) is an integral function, and ...h are positive integers. If, however, f() vanish for za1, a2....where a and limit a, and if for simplicity we assume that z-o is not a zero and all the zeros a, a,.. are of the first order, we find, by applying the preceding theorem to the function () 1df(2), that f(z) = exp [ø(3)] ÏÎ {(1 −2/an) exp on(2)}, dz where +(z) is an integral function, and (2) is an integral polynomial of the form (3) tattsan + The number s may be the 20 same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to take sn. In particular for the function

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F(x) where C is a constant, and г(x) is a function expressible when x is e-11=-1di

real and positive by the integral s

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 b. By taking n large enough, the value of this function, for all positions z of belonging to Ro, differs as little as may be desired from (-a). By taking a sum of terms such as

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we can thus build a rational function differing, in value, in There exist interesting investigations as to the connexion of the Ro, as little as may be desired from a given rational function value of s above, the law of increase of the modulus of the integral f=2A,(t − a) ̄”, function f(z), and the law of increase of the coefficients in the series and differing, outside R or upon the boundary of R, from f, f(z) =Ea, as n increases (see the bibliography below under Integral in the fact that while f is infinite at t=a, F is infinite only at Functions). It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional t=b. By a succession of steps of this kind we thus have the cases, one value at most. For instance, the function exp (z) assumes theorem that, given a rational function of whose poles are every finite value except zero (see below under § 21, Modular outside R or upon the boundary of R, and an arbitrary point c Functions). The two theorems given above, the one, known as Mittag-finite continuous path outside Ŕ from all the poles of the rational outside R or upon the boundary of R, which can be reached by a Leffler's theorem, relating to the expression as a sum of simpler function, we can build another rational function differing in R. functions of a function whose singular points have the point arbitrarily little from the former, whose poles are all at the 818 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, az, as,... 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 a, a,..., and with every point a, there be associated a polynomial in (-a), say g; 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, a pole whereat the expansion consists of the terms g., together with a power series in s-a1; the function is expressible as an infinite series of terms

where y, is also a rational function.

II. With a similar aggregate (a), with limiting points (c), suppose with every point a, there is associated a positive integer r. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order 7, at the point a,, but not elsewhere, expressible in the form

II (1-G-C) "exp (8)

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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= R(1-n-1) exp (i√2.11), when(c) consists of the points of the circle [2]=R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.

89. 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 R, 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 Co. 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 Co; let a, b, be finite points upon C or outside R. Then when b is near enough to a, the fraction (a-b)/(-b) is arbitrarily small for all positions of z; say

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point c.

C and the interior of R can be represented at all points 2 in R by Now any monogenic function f(t) whose region of definition includes f(2) = l-z'

I

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

In particular, to take the simplest case, if Co, C be simple closed polygons, and I 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(2) for all points of R, whose poles are at one finite point c external to г. By a transformation of the form -cr-, 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 4, 62,... to be an indefinitely continued sequence of real positive numbers, converging to zero, and P, to be the polynomial such that, within Co|P,-ƒ(2) 1 <; then the infinite series of polynomials P1(3)+{P2(2) - P1(2)}+{P、(2) − P2(3) } +. . . .

whose sum to n terms is P.(2), converges for all finite values of s and represents f(z) within Co.

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 I of a region, we can suppose the poles of the rational function, constructed to approximate to f(z) within Ro, to be at points of г. A series of rational functions of the form

H1()+{H2(2)-H1(2)}+{H ̧(2)−H2(2)}+.......

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

§ 11. Expression of (1-2)-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 =+in, enclose the points =1, 1+c by means of (i.) the straight lines = a, from 1 to E=1+c, (ii.) a semicircle convex to o of equation (-1)+a, (iii) a semicircle concave too of equation (1−c)2+n2 = a2. The quantities and a are to remain fixed. Take a positive integer so that | is less than unity, and put =(). Now take

G=1+c/r, Gr=1+2c/r,...G=1+c;

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| (1-5)~'~U]<a ̄1{(1+o"i) (1 +0^3)^i (1 +ons)"," :;:;

(1+0) ""1⁄2"). Take an arbitrary real positive ‹, and μ, a positive number, so that <ea, then a value of # such that σ"<μ/(1 +μ) and therefore σ":/(1-σ"1)<μ, and values for m, n,... such that o<o2,

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The rational function U, with a pole at =1+c, differs therefore from (1-3)-1, for all points outside the closed region put about 5=1, 5=1+c, by a quantity numerically less than . So long as a remains the same, and will remain the same, and a less value of e will require at most an increase of the numbers n, m, n, but if a be taken smaller it may be necessary to increase r, and with this the complexity of the function U. εξ Now put (c + 1)3. }= 6+1-5' c+2 thereby the points =0, 1, 1+c become the points 2=0, 1, 0, the function (1-2) being given by (1-2)=c(c+1)~' ( 1 − 3)~1 + (c+1)1; the function U becomes a rational function of 2 with a pole only at =∞0, that is, it becomes a polynomial in z, say, where H is also a polynomial in 2, and

2=

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(x+c)2 + y2 = + C(c+1) y,

a

the points (no, 1−a), (n=0, =1+c+a) become respectively the points (yo, x=c(1-a)/(c+u), (y = o, x=-c(1+c+u)/a), whose limiting positions for ao are respectively (yo, x1), (y = 0, x=-x). The circle (x+c)+ y2=c(c+1)y/a can be written

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where (c+1)/a; its ordinate y, for a given value of x, can therefore be supposed arbitrarily small by taking a sufficiently small. We have thus proved the following result; taking in the plane of z any finite region of which every interior and boundary point is at a finite distance, however short, from the points of the real axis for which Ixo, we can take a quantity a, and hence, with an arbitrary c, determine a number; then corresponding to an arbitrary, we can determine a polynomial P., such that, for all points interior to the region, we have

thus the series of polynomials

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constructed with an arbitrary aggregate of real positive numbers 1, 2, 3,... with zero as their limit, converges uniformly and represents (1-2)- for the whole region considered.

§ 12. Expansion of a Monogenic Function in Polynomials, over a Star Region.-Now consider any monogenic function f(z) of which the origin is not a singular point; joining the origin to any singular point by a straight line, let the part of this straight line, produced beyond the singular point, lying between the singular point and z = ∞, be regarded as a barrier in the plane, the portion of this straight line from the origin to the singular point being erased. Consider next any finite region of the plane, whose boundary points constitute a path of integration, in a sense previously explained, of which every point is at a finite distance greater than zero from each of the barriers before explained; we suppose this region to be such that any line joining the origin to a boundary point, when produced, does not meet the boundary again. For every point x in this region R we can then write

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where f(x) represents a monogenic branch of the function, in case it be not everywhere single valued, and t is on the boundary of the region. Describe now another region R, lying entirely within R, and let x be restricted to be within R, or upon its boundary; then for any point on the boundary of R, the points & of the plane for which st1 is real and positive and equal to or greater than 1, being points for which |2|=|4] or|2|>, are without the region Ro, and not infinitely near to its boundary points. Taking then an arbitrary real positive e we can determine a polynomial in xt-1, say P(xt-1), such that for all points x in Ro we have |(1−xt-1)−P(xt−1)|<e;

the form of this polynomial may be taken the same for all points t of modulus not greater than e, on the boundary of R, and hence, if E be a proper variable quantity

|2=if(x)-(1)P(x-1)= | ƒƒ©E .LM.

where L is the length of the path of integration, the boundary of R, and M is a real positive quantity such that upon this boundary | |t ̃'ƒ(t)|<M. If now P(x)=c+c1x2+. +cx{-",

and

this gives

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|f(x) = {CoμO+C1μix+...+calaix | | = «LM/2#, where the quantities, M, M2, • • are the coefficients in the expansion of f(x) about the origin.

If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points of f(x) to x, it is possible, corresponding to an arbitrary real positive number, to determine a number m, and a polynomial Q(x), of order m, such that for all interior points of this region

f(x)-Q(x) <0.

Hence as before, within this region f(x) can be represented by a series of polynomials, converging uniformly; when f(x) is not a single valued function the series represents one branch of the function.

The same result can be obtained without the use of Cauchy's integral. We explain briefly the character of the proof. If a monogenic function of 1, (1) be capable of expression as a power series in -x about a point x, for 1-xp, and for all points of this circle 1(t)<g, we know that |(^)(x)|<gp ̄"(n!). Hence, taking 12<p, and, for any assigned positive integer, taking m so that for n>m we have (u+n)<(1)", we have

and therefore

where

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ρ

Now draw barriers as before, directed from the origin, joining the singular point of (2) to 2, take a finite region excluding all these barriers, let p be a quantity less than the radii of convergence of all the power series developments of (2) about interior points of this region, so chosen moreover that no circle of radius p with centre at an interior point of the region includes any singular point of(z), let g be such that | $(2)| <g for all circles of radius p whose centres are interior points of the region, and, x being any interior point of the region, choose the positive integer n so that |x|<p; then take the points ax/n, a2 = 2x/n, a;=3x/n.... a. =x; it is supposed that the region is so taken that, whatever x may be, all these are interior points of the region. Then by what has been said, replacing x, 2 respectively by o and x/n, we have

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Σ ... Σ
(x) = 2
λ=0 Ag=0 An=0

(4) (0)

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where hλ++...+λn, K=λ1! Ag!. . . An!, m1 = n2", m2 = n2 m=n', | e| <2g/2

By this formula (x) is represented, with any required degree of accuracy, by a polynomial, within the region in question; and thence can be expressed as before by a series of polynomials converging uniformly (and absolutely) within this region.

§ 13. Application of Cauchy's Theorem to the Determination of Definite Integrals.-Some reference must be made to a method whereby real definite integrals may frequently be evaluated by use of the theorem of the vanishing of the integral of a function of a complex variable round a contour within which the function is single valued and non singular.

We are to evaluate an integral ff(x)dx; we form a closed contour

of which the portion of the real axis from x=a to x=b forms a part,
and consider the integral ff(z)dz round this contour, supposing
that the value of this integral can be determined along the curve
forming the completion of the contour. The contour being supposed
such that, within it, f(z) is a single valued and finite function of the
complex variable 2 save at a finite number of isolated interior points,
the contour integral is equal to the sum of the values of ff(z)dz taken
round these points. Two instances will suffice to explain the
tanxdx is convergent if it be under-
method. (1) The integral
all vanish of the sum of the
stood to mean the limit when e, 5, σ,...
integrals

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S dx, Sus tan xdx,

-e tan x
x

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x

Edx,...

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Finally, for the remaining part of the contour, for which, with R=xx sec α, we have = R(cos +i sin 0) = RE(0), we have exp(-R sin 0)E(iR cos) -exp(R sine)E(iR cos 0) exp(-R sine) E(iR cos 0) +exp(R sin @)E(-R cos @)* when n and therefore R is very large, the limit of this contribution to the contour integral is thus

===ide, i tanz=

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Making a very large the result obtained for the whole contour is

2 √ tan xdx-(x-2a)-2ac=0,

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where is numerically less than unity. Now supposing a to diminish to zero we finally obtain

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(2) For another case, to illustrate a different point, we may take the integral

wherein a is real quantity such that o<a<1, and the contour con-
sists of a small circle, s=E(10), terminated at the points x= cos a,
y= ± sin a, where a is small, of the two lines y sin a for
7 cos axER cos B, where R sin 87 sin a, and finally of a large
circle = RE(14), terminated at the points x = R cos B, y=R sin ß.
We suppose a and ẞ both zero, and that the phase of a is zero for.
rcos aR cos B, y=r sin a R sin 8. Then on r cos axERcos,
y= sin a, the phase of z will be 27, and will be equal to
exp [2ri(a-1)], where x is real and positive. The two straight
R cos 8x-1 dx.
portions of the contour will thus together give a contribution
[1-exp (2ria)] R
rcos a 1+x

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the It can easily be shown that if the limit of zf(2) for z=0 is zero, integral ff(s)dz taken round an arc, of given angle, of a small circle enclosing the origin is ultimately zero when the radius of the circle diminishes to zero, and if the limit of f(s) for so is zero, the same integral taken round an arc, of given angle, of a large circle whose centre is the origin is ultimately zero when the radius of the circle increases indefinitely; in our case with f(x)=24-1/(1+1), we have 2f(2) =2/(1+2), which, for o<a <1, diminishes to zero both for z=0 and for 2=∞. Thus, finally the limit of the contour integral when =0, R=∞ is

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Before passing to this it may be convenient to make here a few Now draw a contour consisting in part of the whole of the positive and negative real axis from x=-n to x = +nx, where n is a positive remarks as to the periodicity of (single valued) monogenic functions. integer, broken by semicircles of small radius whose centres are the To say that f(2) is periodic is to say that there exists a constant w .... the contour containing also the lines such that for every point z of the interior of the region of existence points x, x ==}~,. x= and x=-n for values of y between o and n tan a, where a of f(z) we have f(z+w)=f(z). This involves, considering all existing is a small fixed angle, the contour being completed by the portion periods w=ptio, that there exists a lower limit of p2+ other than zero; for otherwise all the differential coefficients of f(z) would be of a semicircle of radius nr sec a which lies in the upper half of the zero, and f(z) a constant; we can then suppose that not both p plane and is terminated at the points xnx, y=n tan a. Round and are numerically less than e, where >c. Hence, if g be any g) contains only a finite 'tan z dz has the value zero. The contri- real quantity, since the range (-g, this contour the integral number of intervals of length e, and there cannot be two periods w=ptio such that ep (u+1)e, ve=o< (+1)e, where μ, are butions to this contour integral arising from the semicircles of centres −1(25−1)x, +1(25-1), supposed of the same radius, are at once for which both p and are in the interval (-g...g). Considering seen to have a sum which ultimately vanishes when the radius of the integers, it follows that there is only a finite number of periods The part of the contour lying on then all the periods of the function which are real multiples of one semicircles diminishes to zero. tan xdx. The contri-period w, and in particular those periods wherein o<1, there is a lower limit for A, greater than zero, and therefore, since there is the real axis gives what is meant by .2 If bution to the contour integral from the two straight portions at only a finite number of such periods for which the real and imaginary parts both lie between -g and g, a least value of A, say λo. =λow and λ=MA+A', where M is an integer and o<, any x=2x is period Aw is of the form M2+N'w; since, however, . Ma and A are periods, so also is 'w, and hence, by the construction of A. we have 'o, thus all periods which are real multiples of ware expressible in the form M2, where M is an integer, and a period.

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