, region; it will appear that F(z) is also continuous and in fact also a diferenciable funnet non vc=) = S(x, y)ds, for interior or points 20, 2, is a differentiable function of s, having for its differential coefficient the function f (x, y), which Supposing n to be retained the same for all points zo of the region, is therefore also a differentiable function of at interior points. and on to be the upper limit of the possible values of e for the point 2, (3) Hence is the series wo(s) +u(z)+... to co be uniformly conit is to be presumed that so will vary with so, and it is not obvious vergent over a region, its terms being differentiable functions of z. as yet that the lower limit of the values of oo as 2, varies over the then its sum S() is a differentiable function of z, whose differential the condition (2, 20), above, is satisfied for all points z, within or upon series. This theorem, unlike (1), does not hold for functions of a can be divided into a finite number of sub-regions for each of which coefficient, given by zis , is obtainable by differentiating the the boundary of this sub-region, for an appropriate position of zo, within or upon the boundary of this sub-region. This is proved real variable. above as result (B). (4). If the region of definition of a differentiable function fie) Hence it can be proved that, for a differentiable function (), include the region bounded by two concentric circles of radii , R, with centre at the origin, and be an interior point of this region, the integral); }(s)de has the same value by whatever path within the region we pass from 24 to 2. This we prove by showing that when f(x) = 115.10-15.1, where the integrals are both countertaken round a closed path in the region the integral ff(-)dz vanishes. clockwise round the two circumferences respectively: putting in the Consider first a triangle over which the condition, 2) holds, for first (1-2)-1 = $ 20*//**, and in the second (1-2)=-E 1/2001, some position of zo and every position of s, within or upon the boundary of the triangle. Then as we find f(x).- Anco", wherein An-zais puid, taken round any f(3) = f(20)+(2-2)F(x) +90(3-2), wherefol<I, circle, centre the origin, of radius intermediate between r and R. we have Particular cases are: (a) when the region of definition of the $f(z)dz={(z) —20F()\[ds +F(50)/zd8+n10(2-2)dz, function includes the whole interior of the outer circle; then we which, as the path is closed, is nsox:--)dz. Now, from the theorem may take 1 = 0, the coefficients A, for which n o all vanish, and that the absolute value of a sum is less than the sum of the absolute the function {(zo) is expressed for the whole interior (201 <R by a values of the terms, this last is less, in absolute value, than nap: power series 2 Anzo". In other words, about every interior point c of where a is the greatest side of the triangle and p is its perimeter; if the region of definition a differentiable function of 2 is expressible by a A be the area of the triangle, we have A = fab sin C>(ala)ba, where a is the least angle of the triangle, and hence a(a+b+c)<20(6+c) power series in 2-c; a very important result. <451/a; the integral sj(z)ds round the perimeter of the triangle extends to within arbitrary nearness of this on all sides, and at the (B) If the region of definition, though not including the origin, is thus <4794/a. Now consider any region made up of triangles, same time the product 20/() has a finite limit when : diminishes as before explained, in each of which the condition (5, z) holds, as in the triangle just taken. The integral [S(:)dz round the boundary to zero, all the coefficients A, for which n<-m vanish, and we have of the region is equal to the sum of the values of the integral round f(-o) = A 20* + Am+3+1+...+A12-1+A9+A150...to .. the component triangles, and thus less in absolute value than Such a case occurs, for instance, when () = coscc 2, the number m 47K/a, where K is the whole area of the region, and a is the smallest being unity. angle of the component triangles. However small , be taken, such a division of the region into a finite number of component function of 2 is an unclosed aggregate of points, each of which $ 6. Singular Points.—The region of.existence of a differentiable of the region is thus arbitrarily small. Thus it is actually zero, is an interior point of a neighbourhood consisting wholly of which it was desired to prove. Two remarks should be added: points of the aggregate, at every point of which the function is (1) The theorem is proved only on condition that the closed path of definite and finite and possesses a unique finite differential integration belongs to the region at every point of which the conditions are satisfied. (2) The theorem, though proved only when coefficient. Every point of the plane, not belonging to the the region consists of triangles, holds also when the boundary points aggregate, which is a limiting point of points of the aggregate, of the region consist of one or more closed paths, no two of which such, that is, that points of the aggregate lie in every neighbourHence we can deduce the remarkable result that the value of f(z) hood of this, is called a singular point of the function. at any interior point of a region is expressible in terms of the value About every interior point za of the region of existence the function of f(2) at the boundary points. For consider in the original region may be represented by a power scrics in 3-zo, and the series conthe function (3)/(2-2), where zo is an interior point: this satisfies verges and represents the function over any circle centre at a the same conditions as f(e) except in the immediate neighbourhood which contains no singular point in its interior. This has been of . Taking out then from the original region a small := 1/5, polygonal region with zo as centre, the theorem holds for the remain that if the region of existence of the function contains all points of ing portion. Proceeding to the limit when the polygon becomes a the plane for which 1:1>R, then the function is representable for circle, it appears that the integral perfect round the boundary of that the region / existence of the function contains the point ze we? the original region is equal to the same integral taken counterclockwise round a small circle having so as centre: on this circle, the sum of a power series Eanz in : is in absolute value less than M. remain finite for all points z for which l-1 > Ř; for if, for (x1 = R, little from f(20) it is sufficiently small; the value of the integral we have lan!'<Mr", and therefore, if M remains finite for all values round this circle is therefore, ultimately, when r vanishes, equal to if it contains all finite points of the plane cannot contain the point Thus the region of existence of a function arif(). Hence f(2)=zisme 1-20 exp (3) = 23"/n!. This may be regarded as a particular case of a well-known result boundary of the original region. From this it appears that ($ 7), that the circumference of convergence of any power series F(x) = lim representing the function contains at least one singular point. As i dif(e) an extreme case functions exist whose region of existence is circular, there being a singular point in every arc of the circumference, also round the boundary of the original region. This form shows, however small; for instance, this is the case for the functions repre however, that F(20) is a continuous, hnite, differentiable function of sented for 1:1 < by the series 3*, where m=n?, the series :* over the whole interior of the original region. $ 5. Applications. The previous results have manifold appli- where m=n!, and the series 2 */(n+1)(m+2) where m=a*, cations. a being a positive integer, although in the last case the series actually (1) If an infinite series of differentiable functions of be converges for every point of the circle of convergence Izi - 1. 1's uniformly, convergent along a certain path lying with the region be a point interior to the circle of convergence of a series representing of definition of the functions, so that S(z) =(z) +41(s)+...+ the function, the series may be rearranged in powers of 2-20; as 20 Wa-:(z)+R(z), where | R,(2)|<e for all points of the path, we have approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in Ss(a)ds=Sw=>dz+Mw.ce)ds+ +---+ ()dz + StRaceda, + S-m+ ()dz + S+Race)da when , however the isole can be put about the wherein, in absolute value , S, Ra(z)ds <el, if L be the length of the points outside the circle ne convergence of the original series, the path. Thus the series may be integrated, and the resulting series points. If the function be supposed to be given only for the interior is also uniformly convergent. (2) 11 F(x, y) be definite, finite and continuous at every point of a converging beyond the original circle gives what is known as an of the original circle, by the original power series, the scries in 32 region, and over any closed path in the region (x, y)d:=0, then I analytical conlinualion of the function. It appears from what has meet. f(2)-f(x) 2 m., the 20 been proved that the value of the function at all points of its region the sum of the residues at the included poles, a very important result. of existence can be obtained from its value, supposed given by a Any singular point of a sunction which is not a pole is called an series in one original circle, by a succession of such processes of essential singularity; if it be isolated the function is capable, in the analytical continuation. neighbourhood of this point, of approaching arbitrarily near to any $ 7. Monogenic Functions. This suggests an entirely different assigned value.. For, the point being isolated, the function can be way of formulating the fundamental parts of the thcory of represented, in its neighbourhood, as we have proved, by a series functions of a complex variable, which appears to be preferable 3 an(3–4)";itthus cannot remain finite in the immediate neighbour. to that so far followed here. Starting with a convergent power series, say in powers of 3, this hood of the point. The point is necessarily an isolated essential scries can be arranged in powers of s–E, about any point 20 interior singularity also of the function (f(z) – A}"!, for if this were expressible to its circle of convergence, and the new series converges certainly for by a power series about the point, so would also the function (0) be; as (/(z) - Al-' approaches infinity, so does $(2) approach the every position of so this is the greatest radius of convergence of the arbitrary value A. Similar remarks apply to the points=0, the derived series, then the original series represents a function existing hood of an essential singularity, which is a limiting point also of function being regarded as a function of s=2! In the neighbour. only within its circle of convergence. It for some position of % the derived series converges foros-2/<7-17 +D, then it can be poles, the function clearly becomes infinite. For an essential singushown that for points 2, interior to the original circle, lying in the arity which is not isolated the same result does not necessarily hold. annulus r - 18:01 <13–2,<r-121+D, the value represented by the derived series agrees with that represented by the original series. A single valued function is said to be an integral function Il for another point z interior to the original circle the derived series when it has no singular points except s=00. Such is, for converges for 13.-215-11+E, and the two circles 15-2.! - instance, an integral polynomial, which has := 0 for a pole, and beyond 121=, then it can be shown that the values represented by the functions exp (z) which has z=was an essential singularity. these series at these common points agree. Either series then can A function which has no singular points for finite values of be used to furnish an analytical continuation of the function as % other than poles is called a mcromorphic function. If it also originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined have a pole at <= 0 it is a rational function; for then, if by an aggregate of power series of which every one has points of Q1, ...0, be its finite poles, of orders mi, ma, ... convergence common with some one or more others; the whole product (2-0), ...(-0.)"./() is an integral function with aggregate of points of the plane which can be so reached constitutes a pole at infinity, capable therefore, for large values of 2, of an the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have expression (5+1)* (37)”; thus (2–0)" ... (3-2,)"}(5) singular points. The circle of convergence of any of the series has is capable of a form Ebrze, but - E but remains finite for at least one such singular point upon its circumference. So regarded 2=00. Therefore beti=brte= ... =0, and S(:) is a rational the function is called a monogenic function, the epithet having reference to the single origin, by one power series, of the expressions function. representing the function; it is also sometimes called a monogenic If for a single valued function F(z) every singular point in the analytical function, or simply an analylical function's all that is finite part of the plane is isolated there can only be a finite necessary to define it is the value of the function and of all its number of these in any finite part of the plane, and they can be differential cocfficients, at some one point of the planc; in the method taken to be 01. di, as,... with 101110zl laul and limit previously followed here it was necessary to suppose the function laula. About co the function is expressible as A.(2-2.)"; differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, let f.(:) = A":-0.)* be the sum of the negative powers in this that sj(z)da =0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior expansion. Assuming s=o not to be a singular point, let f.(z) be of its circle of convergence. There is another advantage belonging expanded in powers of s, in the form Cri", and to be chosen so to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a that Fo(z) =fo(2) – Cz* = { Caitis, for lal<1.<\ail, less in absolute monogenic function is by no mcans necessarily single valued-it may quite well happen that starting from a particular power series value than the general term c. of a sore-agreed convergent series of converging over a certain circle, and applying the process of analytical real positive terms. Then the series o(s) – F.(3) converges unicontinuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The formly in any finite region of the plane, other than at the points an, notion of basing the theory of functions on the theory of power and is expressible about any point by a power series, and near series is, after Newton, largely due to Lagrange, who has some 2., 6(z)-1.() is expressible by a power series in 2-0.. Thus interesting remarks in this regard at the beginning of his Théorie singularities of F(7) are poles, F(s) is hereby expressed as the sum F(z) – $(z) is an integral function. In particular when all the finite des fonctions analyliques. He applies the idea, however, primarily of an integral function and a series of rational functions. The series is only of very limited validity; for functions of a complex condition. F.(2)!<. is imposed only to render the series EF.(2) variable probably the systematization of the theory owes most to uniformly convergent; this condition may in particular cases be Weierstrass, whose use of the word monogenic is that adopted above, satisfied by a series 2G,(z) where G.(x) = f.(s) – Cos* and v. <us. In what follows we generally suppose this point of view to be regarded as fundamental. An example of the theorem is the function * cot #z-, for which, $ 8. Some Elementary Properties of Single Valued Functions.- taking at first only hall the poles, F.(2)=1/(2-5); in this case the A pole is a singular point of the function /(:) which is not a serics 2F.(z) where F.(2) = (2-5)-++$-is unisormly convergent: singularity of the function 1/[(z); this latter function is therefore, thus 7 cot 62-21- E [(3-5)-4+5-'), where s=o is excluded from by the definition, capable of representation about this point, the summation, is an integral function. It can be proved that this 20. by a series (13)]-1=Eon(3-5.)". If herein a, is not zero we integral function vanishes. can hence derive a representation for {(z) as a power series about Considering an integral function f(a), is there be no finite positions zo, contrary to the hypothesis tbat zo is a singular point for this of z for which this function vanishes, the function (2) is at once seen to be an integral function, 6(2), or f(2) = exp (0(2)]; if however function. Hence do=0; suppose also di=0, Q.=0, ...m=0,great R may be there be only a finite number of values of 2 for which but Qm+0. Then I (3) -1 = (2-2)(am +om+i(z-50)+...), and 1(z) vanishes, say :=Q1, ...om, then it is at once scen that (*) = hence (3.–2.)"(z) = 0m'+Eba(-20)", namely, the expression of exp (=)). (2-2,)... (3 - am)"m, where $(z) is an integral function, (z) about 2=%, contains a finite number of negative powers and ki... who are positive integers. II, however, S(?) vanish for 2 = 21, Az,... where allal... and limit la l, and is for simplicity of 2-20 and a (finite or) infinite number of positive powers. we assume that :-o is not a zero and all the zeros az, ag, .. are Thus a pole is always an isolated singularity. of the first order, we find, by applying the preceding theorem to The integral ${(2)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be arial, where di A, is the coefficient of (2-20)- 'in the expansion of f(x) at the pole: / where $(3) is an integral sunction, and 0.(:) is an integral polynomial the residue of }(?) at the pole. Considering a region in which there of the form *(z) – + +5. The number s may be the are no other singularities than poles, all these being interior points, the integral pari Ss(z)da round the boundary of this region is equal to same for all values of n, or it may increase indefinitely with n: it is sufficient in any case to take son. In particular for the function 1 sin TI sum the rational function of the complex variable i, we have sinta-i { (---) exp (3) }; [ -=>)} in which n is a positive integer, is not infinite at t=0, but has a where n=o is excluded from the product Or again we have pole at (= b. By taking n large enough, the value of this function, 16.-xec. i.{ (1+) exp(-:)) }, for all positions z of 1 belonging to Ro, differs as little as may be desired from (-a)- By taking a sum of terms such as where C is a constant, and r(x) is a function expressible when x is real and positive by the integral so e-ls-de we can thus build a rational function difiering, in value, in Shere 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=EA,(1-2), function (:), and the law of increase of the coefficients in the series and differing, outside R or upon the boundary of R, from f, f(0) = 24,2" as n increases (see the bibliography below under Integral in the fact that while is infinite at 1=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 | 1=b. By a succession of steps of this kind we thus have the cases, one value at most. For instance, the function exp (2) assumes theorem that, given a rational function of t 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 å 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 2=0 as their only limiting point, the other, Weierstrass's factor theorem, giving the expression of an integral function as point c. a product of factors each with only one zero in the finite part ofc and the interior of R can be represented at all points a in Ro by Now any monogenic function (1) whose region of definition includes the plane, may be respectively generalized as follows: I.'If 01, az, 03, ... be an infinite series of isolated points having S(z) - Sindh 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 where the path of integration is C This integral is the limit of a of the points Qı, 22, ..., and with even point a, there be associated a polynomial in (-2,)-, say gi; then there exists a single valued function whose region of existence excludes only the points (a) and Satz (0) (0+1=1), the points (c), having in a point a, a pole whereat the expansion where the points ), are upon C; and the proof we have given of the consists of the terms g., together with a power series in 3-0,; the function is expressible as an infinite series of terms Bryn formly in regard to s, when z is in Ro, so that we can suppose, when existence of the limit shows that the sum S converges to (z) uniwhere y. is also a rational function. II. With a similar aggregate (a), with limiting points (c), suppose far, that the subdivision of C into intervals lit1-6, has been carried sufficiently with every point oi there is associated a positive integer ti.. Then IS-|(2)<< there exists a single valued function whose region of existence excludes only the points (c), vanishing to order 1, at the point a.. for all points = of Ro, where e is arbitrary and agreed upon beforehand. but not elsewhere, expressible in the form The function Sis, however, a rational function of 2 with poles upon C. that is external to Ro. We can thus find a rational function differing (1-4-6) "exp (8), arbitrarily little from S, and therefore arbitrarily little from f(t). for all points z of Ro, with poles at arbitrary positions outside Ra where with every point an is associated a proper point cn of (c), and which can be reached by finite continuous curves lying outside R lan - 6 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 Ho being a properly chosen positive integer. number of sides of C continually greater, we can find a rational If it should happen that the points ( determine a path dividing function differing arbitrarily little from f(z) for all points of Ro whose the plane into separated regions, as, for instance, if an= R(1-n-1) poles are at one finite point c external to r. By' a transformation exp (inv 2.n), when(c) consists of the points of the circle 1:1 = R, the of the form l-c=y-t, with the appropriate change in the rational product expression above denotes different monogenic functions in function, we can suppose this point c to be at infinity, in which case the different regions, not continuable into one another. the rational function becomes a polynomial. Suppose a, 67, : .. 8 9. Construction of a Monogenic Function with a given Region to be an indefinitely continued sequence of real positive numbers, of Existence -A series of isolated points interior to a given converging to zero, and. P, !o be the polynomial such that, within region can be constructed in infinitely many ways whose limiting Co.JP-1(2then the infinite series of polynomials P.(z) +{Pz(z) - P1(z)}+{P:() – P;(z)}+... points are the boundary points of the region, or are boundary whose sum to n terms is P.(z), converges for all finite values of s and points of the region of such denseness that one of them is found represents f(2) within Co. in the neighbourhood of every point of the boundary, however When consists of a series of disconnected polygons, some of small. Then the application of the last enunciated theorem which may include others, and, by increasing indefinitely the number gives rise to a function having no singularities in the interior of of sides of the polygons C, the points C become the boundary points r of a region, we can suppose the poles of the rational function, the region, but having a singularity in a boundary point in every constructed to approximate to f(z) within Re, to be at points of r. small neighbourhood of every boundary point; this function A series of rational functions of the form has the given region as region of existence. H.(:+H2(2)-H.(z)}+{H.(:)-Hz()t... 10 Expression of a Monogenic Function by means of Rational then, as before, represents f() within Ro. And Ro may be taken to Functions in a given Region.-Supposc that we have a region Ro coincide as nearly as desired with the interior of the region bounded of the plane, as previously explained, for all the interior or by r. boundary points of which z is finite, and let its boundary points, $ 11. Expression of (1-2)-1 by means of Polynomials. Appliconsisting of one or more closed polygonal paths, no two of calions.-We pursue the ideas just cursorily explained in some which have a point in common, be called Co. Further suppose further detail. that all the points of this region, including the boundary points, Let c be an arbitrary real positive quantity: putting the comare interior points of another region R, whose boundary is plex variabletin, enclose the points == 1, $=1+c by means of (i.) the straight lines 7= +Q, from = 1 to =1tc, (ii.) a semidenoted by C. Let : be restricted to be within or upon the circle convex to s=o of equation (-1)?+ =a?, (ii.) a semicircle boundary of Co; let a, b, be finite points upon C or outside concave to s=o of equation (1-0)2+12=0?. The quantities R. Then when b is near enough to a, the fraction (2-6)/(2-6) c and e are to remain fixed. Take a positive integer; so that is arbitrarily small for all positions of z; say ) (a) is less than unity, and put --> *). Now take <«, for I a-b1<n; G=I+cz, cz=1+2017, ...c=1 +0; کردی |2rif(x) - S 4 140P(3t+1)| - 1S s«EkoLM, if me, mi...n be positive integers, the rational function constructed with an arbitrary aggregate of real positive numbers 61, 63, 4, . ,, with zero as their limit, converges uniformly and {:-(6})"} represents (1-2)-' for the whole region considered. is finite at $=1, and has a pole of order m at s=c; the rational Star Region. -Now consider any monogenic function f(z) of which $ 12. Expansion of a Monogenic Function in Polynomials, over a function the origin is not a singular point; joining the origin to any singular Ex{1-(**)"}{:-(**)*}" point by a straight line, let the part of this straight line, produced beyond the singular point, lying between the singular point and 2 = 0, is thus finite except for $ = C2, where it has a pole of order nimi be regarded as a barrier in the plane, the portion of this straight line finally, writing from the origin to the singular point being crased. Consider next any finite region of the plane, whose boundary points constitute a COD path of integration, in a sense previously explained, of which every 1-5 point is at a finite distance greater than zero from each of the barriers the rational function before explained; we suppose this region to be such that any line V = (1-5)-(1-x)(1-x2)^(1-43)*1*2. (1-x)*1*?... joining the origin to a boundary point, when produced, docs not has a pole only at S=1+c, of order ning ... tir. meet the boundary again. For every point x in this region R we The difference (1-5)-_U is of the form (1-5)-4P, where P, of can then write the form 1-(1-)(1-P)... (1-pk). Il-x1-10 in which there are equalities among Pl. p... p*, is of the form where f(x) represents a monogenic branch of the function, in case it Epi-Epipa+Epipar-...; therefore, if Ir, lepil, we have be not everywhere single valued, and I is on the boundary of the IPI<Eritenira+Entit. <(1+r) (1+r). (1+)-1; region. Describe now another region Ro lying, entirely within R, and let x be restricted to be within R, or upon its boundary; then now, so long as $ is without the closed curve above described round for any point I on the boundary of R, the points & of the plane for $=1, =Itc, we have which'st is real and positive and equal to or greater than 1, being points for which 121=14| 0121>14), are without the region Ro, and Tiska CCR-1 not infinitely near to its boundary points. Taking then an arbitrary m-5 real positive e we can determine a polynomial in xt-?, say P(xr-'), and hence such that for all points x in Ro we have 1(1-5)-4-U1<a-|(i topi) (1 tons):(1 to*s)".";. 1(1-xt-1)-L-P(xt-)1<e; (i tom,)",";. . •P-). Take an arbitrary real positive e, and H, a positive number, so that the form of this polynomial may be taken the same for all points ? eth-1 <ed, then a value of no such that on; <w/(1+x) and therefore of modulus not greater than e, on the boundary of K, and hence, if E be a proper variable quantity om/(1-onl) <w, and values for na, na, ... such that 09:<FORM, ons nim...om< gps; then, as 1 +x<e', we have 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 (-)--U]<a'{exp (g"1+nơ”:+84ơ" + tnim .Th-1091)-1), 16-TIOL<M. Il now and therefore less than P(x1) = 60+ 6.xl-%+. tcmxp. and w-exp (o": toanit top")-1}, which is less than o [expli** )--] and therefore less than e. 1/(x) - {como tamixt... t Crudeix**|| = ELM/21, The rational function U, with a pole at 5= 1+c, differs therefore where the quantities Ho, Migu... are the coefficients in the exfrom (1-5)for all points outside the closed region put about pansion of S(x) about the origin. $=1, $=1+c, by a quantity numerically less than . So long as I! then an arbitrary finite region be constructed of the kind a remains the same, Yand o will remain the same, and a less value explained, excluding the barriers joining the singular points of f(x) os e will require at most an increase of the numbers ni, ng, ...1: but to x = wo, it is possible, corresponding to an arbitrary real positive if a be taken smaller it may be necessary to increase r, and with this number o, to determine a number m, and a polynomial Q(x), of the complexity of the function U. order m, such that for all interior points of this region V(x)-Q(x) <. Hence as before, within this region f(x) can be represented by a C+1-3 series of polynomials, converging uniformly; when f(x) is not a thereby the points = 0, 1, 1 te become the points :=0, 1, 00, the single valued function the series represents one branch of the function. function (1-0)- being given by (1 - 2)-aceti)-'(1-5)++(6+1)--; The same result can be obtained without the use of Cauchy's the function U becomes a rational function of x with a pole only at integral. We explain briefly the character of the proof. It a s=*, that is, it becomes a polynomial in 2, say#11 - where H monogenic function of l: 0(1) be capable of expression as a power series in 1-* about a point x, for 1-X15p, and for all points of this is also a polynomial in 2, and circle 10(e)]<, we know that 10(x)}<80-*(n!). Hence, taking 121<sp. and, for any assigned positive integer i, taking m so that for > m we have (wtn)*<(1)", we have the lines 7 = * become the two circles expressed, if z=x+iy, by | * (T) 3* |- u ( ( .n12" * and therclore (x+c)?+y=++(6+1), the points (n=0, $= 1-a), (n = 0, $ = 1+c+a) become respectively ON (x+2) = E PP-R 1 Now draw barriers as before, directed from the origin, joining the where -= }{c+1)/a; its ordinate y, for a given value of x, can singular point of $(2) 10 2= , take a finite region excluding all therefore be supposed arbitrarily small by taking a sufficiently small. these barriers, let p be a quantity less than the radii of convergence We have thus proved the following result; taking in the plane of 2 of all the power series developments of (2) about interior points of any finite region of which every interior and boundary point is at a this region, so chosen morcover that no circle of radius p with centre Gnite distance, however short, from the points of the real axis for at an interior point of the region includes any singular point ofo(:). which 1530, we can take a quantity a, and hence, with an !g be such that I o(z)/<g for all circles of radius o whose centres are arbitrary c, determine a number, then corresponding to an arbil interior points of the region, and, x being any interior point of the trary.c, we can determine a polynoinial P., such that, for all points region, choose the positive integer n so that1x1<p; then take the interior to the region, we have |(1-5-) - P.1<c.; points a, = x/n, a: = 2x!n, 0,= 3x/n, ...0. = x; it is supposed that thus the series of polynomials 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, : Pa +(Po-Pu) +(Pr-Pr) + ..., respectively by o and x/n, we have this gives Now put s=(+1)s m 1 2μ 24 త(49) دسم -63d=-=-2a) . we Soo tan « dx=1 Σ ••• 'te, [1-exp (2rio). S. c olm+11)(0) Finally, for the remaining part of the contour, for which, with λι ! ! R=#7 sec @, we have :=R(cos Oti sin 8) = RE(i), we have with dz laml<glaram, side, i tans exp(-R sin 0) E(iR cose) - exp(R sin 0)E(iRcos). provided (wt mit?)*<(I)-,*; in fact for w222-4 it is sufficient exp(-R sino) E(iR cos 8) +exp(R sin 8) EF-i R cosa) to take m=?; by another application of the same inequality, when n and therefore R is very large, the limit of this contribution replacing *, . respectively by G, and xin, we have to the contour integral is thus + BW Making n very large the result obtained for the whole contour is where 18-1<g/pm2ong 28.** ****28-66-22)-2ae=0. provided \tmi+1)<(1) ma+1; we take my=nu? supposing <2724 So long as lg mana and u<2n2 have where e is numerically less than unity. Now supposing a to diminish *+^, <2n2--2, and we can use the previous inequality to substitute to zero we finally obtain here for $(+z) (a). When this is done we find my $1+4, +)(0 (2) For another case, to illustrate a different point, we may take the MON, integral where Lul<2g/wzma, the numbers m, My being respectively man Its Applying then the original inequality to b) (03) = m)(azhin), wherein e is real quantity such that o<a<1, and the contour conand then using the series just obtained, we find a series for $w’(). sists of a small circle, s=r£(10), terminated at the points x=r cosa, This process being continued, we finally obtain y= , sin a, where a is small, of the two lines y= , sin a for 06)(0) * cos aerei cos B, where R sin B=, sin a, and finally of a large circle z=RE(io), terminated at the points x=R cos B, y= =R sin B. AL=0.00 We suppose a and 8 both zero, and that the phase of 2 is zero for where h=2st de toustaa, K=,! dz!... da!, mi = 3n, ma=927-2, ... rcos ax&R cos B, y = y sin a =R sin B. Then on r cos axeR cos , mn=7,141 <2g/2"... y=+ sin a, the phase of s will be 25, and 3e-1 will be equal to By this formula $(x) is represented, with any required degree of -4 exp (2ri(0-1)], where x is real and positive. The two straight accuracy, by a polynomial, within the region in question and portions of the contour will thus together give a contribution thence can be expressed as before by a series of polynomials converging uniformly (and absolutely) within this region. itx" § 13. Application of Couchy's Theorem to the Delerminalion of It can easily be shown that if the limit of 2f(2) for :=o is zero, the Definite Integrals.—Some reference must be made to a method enclosing the origin is ultimately zero when the radius of the circle whereby real definite integrals may frequently be evaluated by diminishes to zero, and if the limit of zf(3) for =is zero, the same use of the theorem of the vanishing of the integral of a function integral taken round an arc, of given angle, of a large circle whose of a complex variable round a contour within which the function centre is the origin is ultimately zero when the radius of the circle is single valued and non singular. increases indefinitely; in our case with f(x) = 80-9/(1+2), we have 25(2) = 2*/(1+2), which, for o çası, diminishes to zero both for z=0 and for s=00. Thus, finally the limit of the contour integral when 1=0, R=is of which the portion of the real axis from x=a to x=b forms a part, and consider the integral S(z)d: round this contour, supposing 1+x that the value of this integral can be determined along the curve Within the contour f(z) is single valued, and has a pole at z=1; at forming the completion of the contour. The contour being supposed this point the phase of z is # and sa-l is exp (17(0-1)) or - exp(iza); such that, within it, f(x) is a single valued and finite ľunction of the this is then the residue of f(2) at 2=-1; we thus have complex variable z save at a finite number of isolated interior points, "Ф ха-1 the contour integral is equal to the sum of the values of Sf(2)d: taken ] dx=-2riexp(ira). I+x round these points. Two instances will suffice to explain the that is method. (1) The integral 18. tandx is convergent if it be understood to mean the limit when , S, 0, ... all vanish of the sum of the $ 14. Doubly Periodic Functions.-An excellent illustration integrals of the preceding principles is furnished by the theory of single ris-e tanx valued functions having in the finite part of the plane no singularities but poles, which have two periods. Now draw a contour consisting in part of the whole of the positive Before passing to this it may be convenient to make here a few and negative real axis from x=-17 to = +17, where n is a positive remarks as to the periodicity on (single valued) monogenic functions. integer, broken by semicircles of small radius whose centres are the To say that (s) is periodic is to say that there exists a constant w points <= ***, * #, ..., the contour containing also the lines such that for every point of the interior of the region of existence ant and x=-nt for values of y between o and nt tan a, where a of (2) we have f(a+w)=f(x). This involves, considering all existing is a small fixed angle, the contour being completed by the portion periods w=ption that there exists a lower limit of p?to other than of a semicircle of radius ar sec a which lies in the upper hall of the zero; for otherwise all the differential coefficients of f(x) would be plane and is terminated at the points * = *#, y=ni tan a. Round zero, and (z) a constant; we can then suppose that not both p this contour the integralstam dz has the value zero. The contri- real quantity, since the range (-8. and o are numerically less than e, where c>c. Hence, if g be any • 8) contains only a finite butions to this contour integral arising from the semicircles of centres number of intervals of leogth e, and there cannot be two periods -|(25-1)*, +}(25-1), supposed of the same radius, are at once integers, it follows that there is only a finite number of periods w=ptio such that we p<(n+1)e, veo<(v+I), where k, v are seen to have a sum which ultimately vanishes when the radius of the for which both p and o are in the interval (-8...8). Considering semicircles diminishes to zero. The part of the contour lying on then all the periods of the function which are real multiples of one the real axis gives what is meant by 2 S**tan i dr. The contri period w, and in particular those periods lw whereino< €1, there is a lower limit for a, greater than zero, and therefore, since there is bution to the contour integral from the two straight portions at only a finite number of such periods for which the real and imaginary x= #nt is parts both lie between -g and g, a least value of 1, say do. If tan iy tan iy N = now and 1= M1+X', where M is an integer ando '<de, any notiy -17 tiy period Aw is of the form Me+Xw; since, however, a. Ma and w are periods, so also is t'w, and hence, by the construction of to, where i tan iy, --sexp (y)-exp(-y)]/exp (y) +exp (-y)), is a real we have X'=0, thus all periods which are real multiples of w are quantity which is numerically less than unity, so that the contri- expressible in the form M2, where M is an integer, and a period. bution in question is numerically less than If beside w the functions have a period w which is not a real ** tana that is than 20. multiple of w, consider all existing periods of the form uw tra' nytgan wherein y, are real, and of these those for which ou1,0 <1; We are to evaluate an integral Sf(x)dx; we form a closed contour pir-s tan dx, dy |