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variables x0, y0, where zosro-Hiyo, over the £: it will appear : F(zo) is also continuous and in fact also a differentiable function ol-fo. Supposing 7 to be retained the same for all points roof the region, and go to be the upper limit of the possible values of e for the point zo, it is to be presumed that go will vary with zo, and it is not obvious as yet that the lower limit of the values of ao as zo 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 (z, zo), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position #: 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 integraffold: has the same value by whatever path within

the region we pass from z1 to z. This we prove by showing that when taken round a closed path in the region the integral f(z)ds vanishes. Consider first a triangle over which the condition (z, zo) holds, for some position of 20 and every position of z, within or upon the boundary of the triangle. Then as

f(z) = f(z)+(z-z)F(x)+"6(2-2), where!6|<1, we have ff(z)dz=[f(z)-z.F(2)]/d2+F(z)/zdz-Hm/9(3-z)ds,

which, as the path is closed, is n/9(z-zo)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 map, where a is the greatest side of the triangle and p is its perimeter; if A be the area of the triangle, we have A = }ab sin C= (a/r)ba, where a is the least angle of the triangle, and hence a(a+b+c) <2a(b+c) <4*Ala; the integral ff(z) dz round the perimeter of the triangle is thus <45 mala. *: consider any region made up of triangles, as before explained, in each of which the condition (z, to) holds, as in the triangle just taken. The integral J f(z)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 4xnk/a, where K is the whole area of the region, and a is the smallest angle of the component triangles. However small y 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 # at the boundary points. For consider in the original region the function f(z)/(z-z), where * is an interior point: this satisfies the same conditions as f(z) except in the immediate neighbourhood of z. Taking out then from the original ion a small regular

»lygonal region with 20 as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon mes a circle, it appears that the integral/# 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 z-zo=rE(ió), dz/(z-zo) = ido, and f(z) differs arbitrarily little from f(zo) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to

- r 2rif(x). Hence f(x)=# boundary of the original region. From this it appears that *

-tim&l=/el--! f# F(z) = lim: ==#--5: (l-zo

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

$5. Applications.—The previous results have manifold applications.

(1) If an infinite series of differentiable functions of z be uniformly convergent, along a certain path lying with the region

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#. where this integral is round the

coefficient.

W(z) -£6. 3)ds, for interior points zo, z, is a differentiable function

of g, having for its differential coefficient the function f(x, y), which is therefore also a differentiable function of z at interior points. (3) Hence if the series us(z)+*(x)+... to do be uniformly convergent over a region, its terms being differentiable functions of 2. then its sum S(z) is a differentiable function of z, whose differential coefficient, given by #f#. is obtainable by differentiating the series. This theorem, unlike (1), does not hold for functions of a real variable. (4) If the region of definition of a differentiable function !' include the region bounded by two concentric circles of radii r, R, with centre at the origin, and zo be an interior point of this region,

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circle, centre the origin, of radius intermediate between r 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 r=o, the coefficients A" for which n <o all vanish, and the function f(z) is expressed for the whole interior |z0|<R by a

power series: Anza". In other words, about every interior point c of

the region of definition a differentiable function of z is expressible by a power series in z-c; a important result. (8) If the region of definition, though not including the origin, extends to within arbitrary nearness of this on all sides, and at the same time the product z"f(z) has a finite limit when |z| diminishes to zero, all the coefficients. An for which n <-m vanish, and we have f(x) =A_mro"+A-meiro"+...+A-1zo"+A9+A1zo...to oo. Such a case occurs, for instance, when f(z) = cosec, z, the number m being unity. $6. Singular Points.-The region of existence of a differentiable function of z is an unclosed aggregate of points, each of which 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 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 zoof the region of existence the function may be represented by a power series in 2-zo, and the series converges and represents the function over any circle centre at ze which contains no singular point in its interior. This has been proved above. And it can be similarly proved, putting 2-1/f, 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 # a power series in 2" or t, in such case we say that the region of existence of the function contains the point z = oo. A series in z* has a finite limit when |z|= oo; a series in 2 cannot remain finite for all points z for which lz|> R; for if, for £ the sum of a power series Xa.z" in z is in absolute value less than M, we have |a. 14 Mr", and therefore, if M remains finite for all values of r however great, an=o. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = co, such is, for instance, the case of the function exp (z) =Xz"/n!. This may be regarded as a particular case of a well-known result (§ 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 2.2", where m = n”, the series 2.2"

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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 a be a point interior to the circle of convergence of a series representing the function, the series may be £ in powers of z-zo; as to #: to a singular point of the function, lying on the circle

convergence, the radii of convergence of these derived series in z--& diminish to zero; when, however, a circle can be put about zo, 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 # these external points. If the function be supposed to be given : for the interior of the original circle, by the original power series, the series inz-zo converging beyond the 9riginal 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. Starting with a convergent power series, say in powers of z, 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 |z-zo|<r-|zo!, if r be the original radius of convergence. If for every position of E: this is the greatest radius of convergence of the £ 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 |z-zol-cr-2|+D, then it can be shown that for points z, interior to the original circle, lying in the annulus r-lzo &Iz-zol−r-|zel-HD, the value £ by the Gerived series agrees with that represented by the original series. If for another point z, interior to the original circle the derived series converges for |z-zıl Ör-|z|+E, and the two circles |2-2) = r-|z2|+D, |z-z,] =r-|z1|+E have interior points common, '' 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 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 £ 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 function ifferentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, that If(z)dz=o 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 monogenicfunctions: 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 a 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 somc interesting remarks in this regard at the beginning of his Théorie des £ 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 : 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(z); this latter function is therefore, by the definition, capable of representation about this point, zo, by a series [f(z)]*=Xa.(2-2)". If herein ao is not zero we can hence derive a representation for f(z) as a power series about zo, contrary to the hypothesis that 20 is a singular point for this function. Hence ao-o; suppose also ai =o, a2=o, ... a.-1=o, but a...+o. Then [f(x)]" =(2-2)"[a-+a-i(-ro)+...], and hence (2-2)"f(z) = a-'-#2b-(z-zo)", namely, the expression of f(z) about z=zo 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 2riAn, where A1 is the coefficient of (z-zo)" 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,

* *we: HJR)is round the wandary of his "tion is ""

the sum of the residues at the included poles, a very important result. Any £ 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.(s-2)";it thuscannot remain finite in the immediate neighbourhood of the point. The point is necessarily an isolated essential singularity also of the function {f(z)-A]", for if this were expressible by a power series about the point, so would also the function £ be; as {f(z)-Al..." approaches infinity, so does f(z) approach the arbitrary value A. Similar remarks apply to the point z = Go, the function being regarded as a function of r=2~3. In the neighbourhood of an essential singularity, which is a limiting point also of £ the function clearly becomes infinite. For an essential singu# which is not isolated the same result does not necessarily old. 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=00 for a pole, and the functions exp (z) which has z=00 as an essential singularity, A function which has no singular points for finite values of 5 other than poles is called a meromorphic function. If it also have a pole at z=00 it is a rational function; for then, if a1, . . . a, be its finite poles, of orders m1, m2, . . . m., the product (2-al)". . . . (z-a.)".f(z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (2')" 2 a,(&")', thus (2-al)". . . . (2-a.)".f(z)

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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) =>a z" 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 MittagLeffler's theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point 2 = 20 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 r £ generalized as follows:

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, there be associated a polynomial in (2-a.)-', say gr; 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 z-ax: the function is expressible as an infinite series of terms gr-yi, where y, is also a rational function.

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

n _* - #) "n ** 1 (' #) exp (ga), where with every point a. is associated a proper point c. of (c), and - # 1. (# ) s *-r-,+,+ U+T). u, being a properly chosen positive integer. "If 'hould appen that the points £etermine a path dividin

the plane into separated regions, as, for instance, if £5 exp (in v 2.n), 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.

§ 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 Ro 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)/(2-b) is arbitrarily small for all positions of 2; say

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we can thus build a rational function differing, in value, in Ro, as little as may be desired from a given rational function J =2A,0t-a)", 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 Ro 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 Ro by I (t)dt f(-)-;###. where the path of integration is C This integral is the limit of a

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where the points t, 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 2, when z is in Ro, so that we can suppose, when the subdivision of C into intervals t, -t, has been carried sufficiently

far, that IS-f(z)| <e, for all points z 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 C, C be simple closed polygons, and T 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 £ for all points of Ro whose poles are at one finite point c external to P. By a transformation of the form t-c =r-", 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 ei, e,, . . . to be an indefinitely continued uence of real positive numbers, converging to zero, and P, to £ polynomial such that, within Co, IP,-f(z) I <e,; then the infinite series of polynomials Pi(z)+{P.(2) – P1(2)]+[P,(2) – P1(2)]+..., whose sum to n terms is P.(2), converges for all finite values of z and re £' 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 P of a region, we can suppose the poles of the rational function, constructed to approximate '' within Ro, to be at points of T. A series of rational functions of the form Hi(z)+(H1(2)-H1(2)]+[H,(2)-H1(2)]+... then, as before, represents f(z) within R. And R2 may be taken to £e as nearly as desired with the interior of the region bounded y T. § 11. Expression of (1-z)- 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 t=# 4-in, enclose the points i = 1, r = 1 +c by means of (i.) the straight lines n = *a, from # = 1 to # = 1 + c, (ii.) a semicircle convex to t =o of equation (5-1)*-i-n + a”, 6: a semicircle concave to t =o of equation (8-1-4)*-i- m = a”. The quantities c and a are to remain fixed. Take a positive integer r so that

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constructed with an arbitrary aggregate of real positive numbers *1, e2, es, ... with zero as their limit, converges uniformly and represents (1-z)~" 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 int by a straight line, let the part of this straight line, produced £ the £ point, lying between the singular point and z = oo, 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 int is at a finite distance greater than zero from each of the barriers £ explained; we suppose this region to be such that any line joining the £ to a boundary point, when produced, meet the boun can then write

- does not ary again. For every point x in this region R we

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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 Ro lying, entirely, within R. and let x be restricted to be within Ro or upon its boundary; then for any point t on the boundary of R, the points z of the plane for which st" is real and positive and equal to or greater than 1, bein points for which |z|=|t] oriz|>|t|, are without the region Re, £ not infinitely near to its boundary points. Taking then an arbitrary real positive e we can determine a polynomial in xt", say P(xt"). such that for all points x in Rowe have

1(1-xt-:)--P(xt-1)|<e; the form of this polynomial may be taken the same for all points t on the boundary of #" and hence, if E be a proper variable quantity of modulus not greater than e,

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| ** |<+ # #. < :-
p"..., 1*" **

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 402) about interior points of this region, so chosen moreover that no circle of radius e with centre at an interior point of the region includes any singular point of s(z). let g be such that I p(z)|<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: xl & le; then take the points al = x/n, a2-2xln, 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, a respectively by o and x/n, we have

where

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Applying then the original inequality to 4')(a)=4"(*#x/n), and then using the series just obtained, we find a series for 4"(as). This process being continued, we finally obtain

* * fx. A. *::...: '' ()'t. A1*0 A2 =0 A**0

where h =A1+A2+...+\a, K=\ll \s!... A.!, m1=n", ma="...., m. =n", le 1 <2g/2".

By this formula $(x) is represented, with any required. : of accuracy, by a polynomial, within the region in question; and thence can #. 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 issingle valued and nonsingular.

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Now draw a contour consisting in part of the whole of the positive and negative real axis from x=-n" to x = + nr., where n is a positive integer, broken by semicircles of small radius whose centres are the points x = ***, x = ***, ... , the contour containing also the lines * =nr and x=-nx for values of y between o and nx tana, where a is a small fixed angle, the contour £ completed by the portion of a semicircle of radius nr sec a which lies in the upper half of the flane and is terminated at the points x = *nr, y=nr tan a Round this contour the integral #. has the value zero. The contributions to this contour integral arising from the semicircles of centres -*(2s-1)*, +1(2s-1)*, sup of the same radius, are at once seen to have a sum which £ vanishes when the radius of the semicircles diminishes to zero. The part of the contour lying on

- - - - **tan x the real axis gives what is meant by. J. +ax. The contri

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§ 14. Doubly Periodic Functions.—An excellent illustration of the preceding principles is furnished by the theory of single valued functions having in the finite part of the plane no singularities but poles, which have two periods.

Before passing to this it may be convehient to make here a few remarks as to the periodicity '' valued) monogenic functions. To say that f(z) is periodic is to say that there exists a constant to such that for every point z of the interior of the region of existence of f(z) we have f(z+k)=f(e). This involves, considering all existing periods w = p +io, that there exists a lower limit of p24-g" other than zero; for otherwise all the differential coefficients of f(z) would be zero, and f(z) a constant; we can then suppose that not both p and a are numerically less than e, where e >c. Hence, if g be any real quantity, since the range (-g, ... g) contains only a finite number of intervals of i£ e, and there cannot two periods w=e-Hia such that we'e 3 (a+1)e, veea 4 (v-H1)e, where u, v are integers, it follows that there is only a finite number of periods for which both p and * are in the interval (-g .. .g). Considering then all the periods of the function which are real multiples of one period w, and in particular those periods Aw whereino ~A*1, there is a lower limit for A, greater than zero, and therefore, since there only a finite number of such periods for which the real and £ parts both lie between-g and g, a least value of A, say No. If in=A* and A-MA2+A', where is an integer and oSA'4x2, a period Aw is of the form Mo-HA'o'; since, however, Q, Mo and N are periods, so also is \'o, and hence, by the construction of No. we have X =o, thus all periods which are real multiples of a expressible in the form Mo, where M is an integer, and 9 a £:

If beside a the functions have a period to which is not a real multiple of w, consider all existing periods of the ####. wherein u, v are real, and of these those for which deper?'

that is

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