I variables to yo, where 20=tiyo over the region; it will appear +(z) = that F(x) is also continuous and in fact also a differentiable function Silla, y)da, for interior points 20 , s, is a differentiable function of , having for its differential coefficient the function f (x, y), which Supposing to be retained the same for all points zo of the region, is therefore also a differentiable function of z at interior points. and to be the upper limit of the possible values of e for the point 20, (3) Hence is the series wo(z) +41(2)+... to be uniformly conit is to be presumed that so will vary with zo, and it is not obvious vergent over a region, its terms being differentiable functions of 2, as yet that the lower limit of the values of oo as 2. varies over the then its sum S(z) is a differentiable function of s, whose differential region may not be zero. i S(1)d! can be divided into a finite number of sub-regions for each of which coefficient , given by is obtainable by differentiating the the condition (3,2), above, is satisfied for all points 2, within or upon series. This theorem, unlike (1), does not hold for functions of a the boundary of this sub-region, for an appropriate position of so, 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 f(x) Hence it can be proved that, for a differentiable function S(2), 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, the integral S(e)ds has the same value by whatever path within $(30) = Stereo Ishod, where the integrals are both counterthe region we pass from 2: to s. This we prove by showing that when 2riR-20 2ri. taken round a closed path in the region the integral ff(s)dz vanishes.clockwise round the two circumferences respectively; putting in the Consider first a triangle over which the condition (2, 4) holds, for first (1-2)-1= € 205/+1, and in the second (1-2)=-/200+1, some position of % and every position of 2, within or upon the boundary of the triangle. Then as we find f(2)= 2 Anto", wherein A.-de, taken round any f(*)=f(20)+(3->)F(x) +90(3-2), where101<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 1/(:)d2=[(*)-20F(20)/dz+F(-o) Sad: +110(3-2)dz, function includes the whole interior of the outer circle; then we which, as the path is closed, is nfo(z-2)dz. Now, from the theorem may take =o, 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 f(zo) is expressed for the whole interior 1201 <R by a values of the terms, this last is less, in absolute value, than nap: power series & Azzo". In other words, aboul every interior point c of where e is the greatest side of the triangle and p is its perimeter; if A be the area of the triangle, we have 4 - fab sin C>(al*)ba, where the region of definition a differentiable function of is expressible by a s is the least angle of the triangle, and hence a(a +6+4) <22(6+c) power series in 2-c; a very important result. <ørA/a; the integral Sy(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 <474/2. Now consider any region made up of triangles, same time the product z-|(:) has a finite limit when 1:1 diminishes as before explained, in each of which the condition (s, zo) holds, as to zero, all the coefficients A, for which n<-m vanish, and we have in the triangle just taken. The integral [[(a)dz round the boundary of the region is equal to the sum of the values of the integral round $(30)=A mio +A_m+180#*++...+A134++A120...to c. the component triangles, and thus less in absolute value than Such a case occurs, for instance, when /()= cosec 2, the number m 47K/e, 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 : is an unclosed aggregate of points, each of which $ 6. Singular Points.—The region of.existence of a differentiable triangles has been shown possible; the integral round the perimeter 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 ditions 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 ed paths, no two of whi such, that is, that points of the aggregate lie in every neighbourHence, we can deduce the remarkable result that the value of f(e) hood of this, is called a singular point of the function. About every interior point zo of the region of existence the function of) at the boundary points. For consider in the original region may be represented by a power series in 3-2, and the series conthe function f(:)/(-u), where zo is an interior point: this satisfies verges and represents the function over any circle centre at 20 the same conditions as f(2) except in the immediate neighbourhood which contains no singular point in its interior. This has been is. Taking out then from the original region a small regular that if the region of existence of the function contains all points of And it can be similarly proved, putting 2=1/5, paygonal region with so as centre, the theorem holds for the remaining 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 s dzf(2) round the boundary of that the region of existence of the function contains the point := ? the original region is equal to the same integral taken counter A series in has a finite limit when 1:1 = 0; a series in 2 cannot dockwise round a small circle having zo as centre; on this circle, the sum of a power series Eon7" in z is in absolute value less than M. remain finite for all points x for which 1:1 > R; for if, for 121 = R, bever, if :-***E(8), dz!(:-) = ido, and f(z) differs arbitrarily Irtle from fise) if y is sufficiently small; the value of the integral of r however great.&n=0. Thus the region of existence of a function we have la l< Mr, and therefore, if M remains finite for all values found this circle is therefore, ultimately, when , vanishes, equal to if it contains all finite points of the plane cannot contain the point zifa). Hence f(x) = sing where this integral is round the such is, for instance, the case of the function exp (z) = 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 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, also mend the boundary of the original region. This form shows, however small; for instance, this is the case for the functions repre however, that F(x) is a continuous, finite, differentiable function of 20 sented for 181 < 1 by the series 2, where m=n’, the series 3 * Grer the whole interior of the original region. 5. Applications.—The previous results have manifold appli- where m=n!, and the series, 3 3*/(m+1)(m+2) where m=2", cations. a being a positive integer, although in the last case the series actually (1) If an infinite series of differentiable functions of z be converges for every point of the circle of convergence 121=1. 113 Siformly convergent along a certain path lying with the region be a point interior to the circle of convergence of a series representing al debastion of the functions, so that S(x) = 10(x) +1(3)+...+ the function, the series may be rearranged in powers of :--.; as zo While+ R.(s), where | Ra(z) | <e for all points of the path, we have approaches to a singular point of the function, lying on the circle Ssds-S(a)ds+fw.ce)ds+ . +S*-- (2)dz + SRace)de , mendimin to zero; when, however, a circle can be put about som wherein, in absolute value , s'R.(ə)dz<el, if I be the length of the points putside the circle of 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. (3) Li sex, 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 series in : region, and over any closed path in the region (x, y)dz=0, then I analytical continuation of the function. It appears, from what has raif(t) F(x) = lim. (2)-f(20) ! m., the ro 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 function 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 87. 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 theory of represented, in its neighbourhood, as we have proved, by a series functions of a complex variable, which appears to be preferable 3 an(2-zo)";it thus cannot remain finite in the immediate neighbourto that so far followed here. Starting with a convergent power series, say in powers of z, this hood of the point. The point is necessarily an isolated essential series can be arranged in powers of 2–3, about any point zo interior singularity also of the function (f(s) - 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 f(z) 13-2<-- 1201, if, be the original radius of convergence. Il for be; as (S() -Al approaches infinity, so does f(z) approach the every position of so this is the greatest radius of convergence of the arbitrary value A. Similar remarks apply to the point : = 4, the derived series, then the original series represents a function existing hood of an essential singularity, which is a limiting point also of the derived series converges for 15-20 <1-12 +D, then it can be poles, the function clearly becomes infinite. For an essential singushown that for points z, interior to the original circle, lying in the arity which is not isolated the same result does not necessarily hold. annulus 1-1201 <12-20 <1-1201+D, the value represented by the derived series agrees with that represented by the original scries. A single valued function is said to be an integral function Il sor another point a interior to the original circle the derived series when it has no singular points except :=00. Such is, for converges for 15-21<r-13:1+E, and the two circles 12-2) = instance, an integral polynomial, which has z=00 for a pole, and beyond 121=r, then it can be shown that the values represented by the functions exp () which has 2 = C as an essential singularity. these scries 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 s other than poles is called a meromorphic 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 z=00 it is a rational function; for then, if by an aggregate of power series of which every one has points of 21, : : 9, be its finite poles, of orders mi, ma, ... convergence common with some one or more others; the whole product (3-0)", ...(-0.)" /(z) 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 region are the points in whose neighbourhood the derived series have expression (2-0)-* £ 4.(-1)"; thus (3-0)", ...(-0.)-MO) radii of convergence diminishing indefinitely to zero; these are the is capable of a form E biz", but - E best remains finite for 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 refer- :=. Therefore br+1 = br+2= =o, and S(:) is a rational ence 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(x) every singular point in the analytical function, or simply an analytical functions 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 coefficients, at some one point of the planc; in the method taken to be G10z, as. . . . with local laal ... and limit previously followed here it was necessary to suppose the function lan=co. About or the function is expressible as A.(-a)"; differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, let fa(z) = È A"(2-0.)" be the sum of the negative powers in this that Sf(z)dz=0 over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior expansion. Assuming 2 = 0 not to be a singular point, let folo) be of its circle of convergence. There is another advantage belonging expanded in powers of z, in the form & Caze, and we 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 F.(z) – f.(z) – E Coz* = * Cozris, for s}<>,<lc.), less in absolute monogenic function is by no means necessarily single valued—it may quite well happen that starting from a particular power series, value than the general term e, of a fore-agreed convergent series of converging over a certain circle, and applying the process of analytical real positive terms. Then the series 0(2) = { F.(2) converges unithe value obtained does not agree with the initial value. The formly in any finite region of the plane, other than at the points en notion of basing the theory of functions on the theory of power and is expressible about any point by a power series, and near Thus series is, alter Newton, largely due to Lagrange, who has some Q., $(2) -\.() is expressible by a power series in s-a.. interesting remarks in this regard at the beginning of his Théorie F(z) – (2) is an integral function. In particular when all the finite des fonctions analytiques. He applies the idea, however, primarily singularities of F(s) are poles, F() is hereby expressed as the sum to functions of a real variable for which the expression by power 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)| <e is imposed only to render the series F.:) 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 G.() where G,(z) = f.(z) – Crik and ». <us In what follows we generally suppose this point of view to be regarded as fundamental. An example of the theorem is the function scot -r for which, $ 8. Some Elementary Properties of Single Valued Functions.- taking at first only half the poles, f.(z) = 1/(2-5); in this case the A pole is a singular point of the function (s) which is not a serics EF.(z) where F.(z) = (3-5)-175-4 is uniformly convergent: singularity of the function 11/(z); this latter function is therefore, thus a cot 15-51-3 (:-s)-4+5-1], 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 ((:)] i=0,(2–2.)*. If hercin Q, is not zero we integral function vanishes. can hence derive a representation for s(z) as a power series about Considering an integral function |(+), if there be no finite positions zo, contrary to the hypothesis that zo is a singular point for this of = for which this function vanishes, the function All (o) is at once seen to be an integral function, o(?), or S(x) = exp (0)]; if however function. Hence 0o=0; suppose also a;=0, 02=0, ...0m-1=0, great R may be there be only a finite number of values of = for which but ao. Then || ()]= (-20)"(6mtom+1(3-20)+...), and 1(3) vanishes, say := 21. ...am, then it is at once seen that f(e) = hence (3.–2.)-|(z) =a '+Ebn12-20)", namely, the expression of exp ($(=)). (2–0,541... (2-am)n, where o(s) is an integral function, (2) about :=zo contains a finite number of negative powers and has .hm are positive integers. 11. however, f(z) vanish for: *9. ag, . . . where la.15 10:15 ... and limit laxl=, and if for simplicity of :-30 and a (finite or) infinite number of positive powers. we assume that :-o is not a zero and all the zeros az, az, ., are Thus a pole is always an isolated singularity. of the first order, we find, by applying the preceding theorem to The integral Sf(z)dz taken by a closed circuit about the pole not 0fc), that f(3) = exp{(2)) i {(1-z/en) exp $1(E)). containing any other singularity is at once seen to be aria, where A, is the coefficient of (2-2)- in the expansion of f(x) at the pole; where $(z) is an integral function, and on(2) is an integral polynomial 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 of the form on (2) + The number s may be the are no other singularities than poles, all these being interior points, same for all values of n, it the integral i SS(=)dround the boundary of this region is equal to may increase indefinitely with ~; it is sufficient in any case to take san. In particular for the function 1 1 the function to de so sn ss, we have 21 sum the rational function of the complex variable i, intr-fi { (---) exp () }; [-(=%] in which n is a positive integer, is not infinite at 1-8, but has a where :=o is excluded from the product Or again we have pole at i = b. By taking n large enough, the value of this function, = sec, i,{(1+) exp(-:)) }; for all positions : of 1 belonging to Re, differs as little as may be desired from (2-a)- By taking a sum of terms such as where C is a constant, and (x) is a function expressible when x is real and positive by the integral ral e-le-ldt 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 s = EA,(1-2)-P, function f(e), and the law of increase of the coefficients in the series and differing, outside R or upon the boundary of R, from S, 110) = 20.5 as n increases (see the bibliography below under Integral in the fact that while s is infinite at 1=0, 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 1 whose poles are every finite value except zero (see below under às 21, Modular outside R or upon the boundary of R, and an arbitrary point o functions). The two theorems given above, the one, known as Mittag- finite continuous path outside Ř from all the poles of the rational outside R or upon the boundary of R, which can be reached by a Leller's theorem, relating to the expression as a sum of simpler function, we can build another rational function differing in Ro functions of a function whose singular points have the point arbitrarily little from the former, whose poles are all at the s=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 of c and the interior of R can be represented at all points : in Ro by Now any monogenic function f(t) whose region of definition includes the plane, may be respectively generalized as follows:I'll Go Gs, 02 ... be an infinite series of isolated points having f(2)= ([()d1 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 01, 02, ..., and with every point a, there be associated a polynomial in (s-a.)-, say gii then there exists a single valued function whose region of existence excludes only the points (a) and Sz4, the points (c), having in a point Qi 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 by formly in regard to z, when : is in Ro, so that we can suppose, when existence of the limit shows that the sum S converges to ]() uniwhere y, is also a rational function. II. With a similar aggregate (c), with limiting points (c), suppose far, that the subdivision of C into intervals biti-h, has been carried sufficiently th every point oi there is associated a positive integet ri. Then 15-4(:)136, there exists a single valued function whose region of existence excludes only the points (c), vanishing to order ?, at the point Qi, for all points . of Ro, where e is arbitrary and agreed upon beforehand. but oot elsewhere, expressible in the form The function Sis, however, a rational function of z with poles upon C, that is external to Ro. We can thus find a rational function differing îi (1-6--6) *exp(80) arbitrarily little from S, and therefore arbitrarily little from f(t), for all points z of Ro, with poles at arbitrary positions outside Re where with every point on is associated a proper point ca of (c), and which can be reached by finite continuous curves lying outside R from the points of C., 8.-:-) In particular, to take the simplest case, if Co, C be simple closed polygons, and r be a path to which C approximates by taking the pe 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 (c) determine a path dividing function differing arbitrarily little from f(a) for all points of Ro whose the plane into separated regions, as, for instance, if an = R(1-1-1) poles are at one finite point ( external to r. By a transformation exp is v2.n), when(c) consists of the points of the circle 1:1 = R, the of the form 1-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 €, , § 9. Construction of a Monogenic Function with a given Region to be an indefinitely continued sequence of real positive numbers, Existence “A series of isolated points interior to a given converging to zero, and. P. to be the polynomial such that, within Co, Pr-S(7) I <; then the infinite series of polynomials region can be constructed in infinitely many ways whose limiting P.(8) +{Pz(z) – P. (2)}+{P.(z) – P2 (2)}+..., points are the boundary points of the region, or are boundary whose sum to » terms is P.(2), converges for all finite values of z 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 C 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 y(z) within Ro, to be at points of r. all neighbourhood of every boundary point; this function A series of rational functions of the form has the given region as region of existence. H.(z)+{Hz(2)-H (2)}+{H,(6)-Hz()}+... 10 Expression of a Monogenic Function by means of Rational then, as before, represents f(z) within Ro. And R, may be taken to Functions in a given Region.-Suppose 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)' by means of Polynomials. Appliconsisting of one or more closed polygonal paths, no iwo of cations.-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 variable in enclose the points == 1, s=1+c by means denoted by C. Let z be restricted to be within or upon the circle convex to s = of equation (4-1)?+? = a, (iii) a semicircle boundary of Co; let a, b, .. be finite points upon C or outside concave to ;=o of equation (41-ci+m=a?. The quantities R. Then when is near enough to a, the fraction (0-6)/(2-6) and a are to remain fixed. Take a positive integer 1 so that is arbitrarily small for all positions of z; say is less than unity, and put on =;(). Now take <e for 10-01<; 6=Iter, 6=1+2cr, ...(=1+e; 1 "{:-(8=1)"} if m, na, ... Mhp be positive integers, the rational function constructed with an arbitrary aggregate of real positive numbers 61, 62, 63, ... with zero as their limit, converges uniformly and I-5 represents (1-2)' for the whole region considered. 12. Expansion of a Monogenic Function in Polynomials, over a is finite at S=1, and has a pole of order ni at S=C; the rational Slar Region.--Now.consider any monogenic function f(s) of which function 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:=, is thus finite except for S=C, where it has a pole of order nina; be regarded as a barrier in the plane, the portion of this straight line finally, writing 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 the rational function before explained; we suppose this region to be such that any line U = (1-5)-(1-x)(1-x2)*(1-43)*1* . (1-3)*1y.,.ro joining the origin to a boundary point, when produced, does not has a pole only at s= !+c, of order nim ...the. meet the boundary again. For every point x in this region R we The difference (1-5) -L-U is of the form (1-5)-P, where P, of can then write the form de fl) 1-(!-2)(1-2)... (1-px), 2 rif(x) 1 1-xf-1. in which there are equalities among P, A,... Pa, is of the form where f(x) represents a monogenic branch of the function, in case it Ep-Epipa+Epipps ...; be not everywhere single valued, and I is on the boundary of the therefore, if 10.1=lil, we have region. Describe now another region Ro lying, entirely within R, IPI<Eri +Erivat Eryritit . <(it) (1 tra).. (1+r)-1; 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, $=Itc, we have which z-i is real and positive and equal to or greater than 1, being Cm-m-1 points for which 121=14] orl s1>14), are without the region Re, and not infinitely near to its boundary points. Taking then an arbitrary and hence real positive e we can determine a polynomial in x?-?, say P(xt-'). such that for all points x in Ro we have |(1-5)---Ul<«-|(1 toni) (i tomz)"}(i tors)","s; 1(1-xt-1)-2 P(xt-')[<c; (1 +o",)",">..-11. Take an arbitrary real positive «, and H, a positive number, so that the form of this polynomial may be taken the same for all points CM-1 <ca, then a value of n such that on <w/(1+x) and therefore of modulus not greater than , on the boundary of R, and hence, if E be a proper variable quantity o*i/(1-0") <, and values for na, naro. such that ocno Ons sasni,..on O""; then, as 1+x<e, we have where L is the length of the path of integration, the boundary of RA 11,12 1 ...t-1 and M is a real positive quantity such that upon this boundary 1(-5)=4-U1<a-h{exp (oni tn,gest hinzonot... trim so Mr-1099)-1}, TO<M. Il now and therefore less than P(xt) = co tcxt%t. +3, and alexp (oni talnit torps)-1), which is less than OM this gives and therefore less than e. 1$() - {cowot cwx+ ... + Continuitet |= CLM/27, The rational function U, with a pole at s=1+c, differs therefore where the quantities ho. H... are the coefficients in the exfrom (1-5), for all points outside the closed region put about pansion of f(x) about the origin. $= 1, S = 1 +ć, by a quantity numerically less than e. So long as I! then an arbitrary finite region be constructed of the kind a remains the same, i and o will remain the same, and a less value explained, excluding the barriers joining the singular points of f(x) of e will require at most an increase of the numbers ni, M, ...; but to x = 0, it is possible, corresponding to an arbitrary real positive if a be taken smaller may be necessary to increase 1, 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 (1)-Q(x) < 05 Now put Hence as before, within this region f(x) can be represented by a 6+1-3 series of polynomials, converging uniformly; when f(x) is not a thereby the points $=0, 1, 1+c become the points s=0, 1, , the single valued function the series represents one branch of the function. function (1-2)-! being given by (1 -=--=c(+1)+(1-5)-+*c+1)--; The same result can be obtained without the use of Cauchy's the function U becomes a rational function of a with a pole only at integral. We explain briefly the character of the proof. If a s=, that is, it becomes a polynomial in z, say+-, where H going on in function au point to be capable of expression as a power is also a polynomial in s, and circle 10(1)]<s, we know that lon (x)!<80*(n!). Hence, taking 131< tp, and, for any assigned positive integer M, taking m so that for n>m we have (u + n) <(1)", we bave C+1 the lines y = + become the two circles expressed, if z=rtiy, by (+ n)! and therefore 'se (a) the points (n=0,6 = 1-2), (n=0,&=1tcta) become respectively (w)(x+2) – E 29 teme PM - 1 Now draw barriers as before, directed from the origin, joining the where y = {c(+1)/a; its ordinate y, for a given value of x, can singular point of $(z) to := 0, 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 3 of all the power series developments of o() about interior points of any finite region of which every interior and boundary point is at a this region, so chosen moreover that no circle of radius p with centre finite distance, however short, from the points of the real axis for at an interior point of the region includes any singular point of $(z), which 1x5, we can take a quantity e, and hence, with an let g be such that l (2)| <g for all circles of radius p whose centres are arbitrary c, determine a number r; then corresponding to an arbi interior points of the region, and, x being any interior point of the trary c. we can determine a polynomial P., such that, for all points region, choose the positive integer n so that 1x1<}p; then take the interior to the region, we have |(1-z-1) - P.I<ci points Q1 = x/n, 4= 2x/n, Q;= 38/n, ...Q.=r; it is supposed that the region is so taken that, whatever x may be, all these are interior thus the series of polynomials points of the region. Then by what has been said, replacing x, s Ps+(Po-P.)+(Po-P:)+ ..., respectively by o and x/n, we have (x+c)?+y = +"(+1) (x+c)?/(x+c)*+v14(x+c)]} 2 2μ we Σ... Σ ()*+ Finally, for the remaining part of the contour, for which, with Aq=0 RENT sec a, we have 3=R(cos Oti sin 0)=RE(i), we have with ds laul<glazmi, sido, i tan s= exp(-R sin 8) E(iR cos) - exp(R sin e)E(iRcos I) exp(-R sino) ECR cos 8) +exp(R sin e)ET-i cos o): provided (wtmi+1)*<(1)m, t; in fact for y2n2-3 it is sufficient to take mamla; by another application of the same inequality, when n and therefore R is very large, the limit of this contribution replacing #, respectively by Q and xin, we have to the contour integral is thus ou (* 8m do =-(0-20). φα=Σ' Making a very large the result obtained for the whole contour is where 18'<g/pu2mg provided witma+1)* <(1)m+1; we take me = nbus, supposing #<2n** So long as ng manin? and » <anard have where e is numerically less than unity. Now supposing a to diminish the <2ni-7, and we can use the previous inequality to substitute to zero we finally obtain here for $6+y)(Q). When this is done we find tan *) (2) For another case, to illustrate a different point, we may take the Hy=O4, 0 integral where 4.Bul<2g/pm2my, the numbers m, me being respectively mzdo Applying then the original inequality to do) (0:0) hem) wherein e is real quantity such that o<a <!; and the contour conand then using the series just obtained, we find a series for $(6.). sists of a small circle, z=r£(10), terminated at the points x=r cos a, This process being continued, we finally obtain y= *, sina, where a is small, of the two lines y= , sin a for i cos axi cos B, where R sin B=r sin a, and finally of a large Φ(α) – Σ. K circle 2=RE(io), terminated at the points x=Rcos B, y= =R sin 8. do ng Odno We suppose a and B both zero, and that the phase of s is zero for where kalitigt.ustha, K=1,! dy!...da!, me = 720, ms=320-4, rcos adR cos B. y=sin a =R sin B. Then on, cos a ExERcos B. A =9,1 € 1 <2g/2".. y = sin a, the phase of s will be 27, and 24-1 will be equal to By this formula $(x) is represented, with any required degree of ** exp (ri(0-1)], where 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 con [l-exp (2ric)] Somos -dx. verging uniformly (and absolutely) within this region. cosa 1+* $ 13. Application of Cauchy's Theorem lo the Determination of It can easily be shown that if the limit of 2f(a) for s=o is zero, the Dejnite Integrals.-Some reference must be made to a method integral s1(c)ds taken round an arc, of given angle, of a small circle 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 ef(3) for 2= 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 /(?),= 80-1/(1+z), we have zf() = 8°/(1+x), which, for osa<1, diminishes to zero both for s=0 We are to evaluate an integral $$(x)dx; we form a closed contour and for s=00. Thus, finally the limit of the contour integral when r=0, R = is of which the portion of the real axis from x=a to x=b forms a part, (1-exp (2ria) and consider the integral [f(z)dz round this contour, supposing 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 s=1; at forming the completion of the contour. The contour being supposed this point the phase of z is r and 3-1 is exp (ir(2-1)] or - exp(ira); such that, within it, $(z) is a single valued and finite function of the this is then the residue of f(z) at <=-1; we thus have complex variable : save at a finite number of isolated interior points, rooma-1 the coatour integral is equal to the sum of the values of ff(s)da taken [1-exp(aria)] dx=2riexpira), 1+x round these points. Two instances will suffice to explain the that is 07-1 method. (1) The integral s tan &de is convergent if it be under _dx=r cosec (or). stood to mean the limit when 4, 5, 6, ... all vanish of the sum of the $ 14. Doubly Periodic Funclions.-An excellent illustration integrals of the preceding principles is furnished by the theory of single - tani valued functions having in the finite part of the plane no singularities but poles, whích 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--no to x= +17, 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(z) is periodic is to say that there exists a constant w points = + 3*,*=, ..., the contour containing also the lines such that for every point s of the interior of the region of existence 1=nr and x=-ns for values of y between o and nz tan a, where a of f(z) we have f(x+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 pl to other than of a semicircle of radius nt sec a which lies in the upper hall of the zero; for otherwise all the differential coefficients of f(z) would be plane and is terminated at the points x= #na, y=nt tan a. Round zero, and f(x) a constant; we can then suppose that not both p tan and o are numerically less than , where e >c. Hence, if g be any butions to this contour integral arising from the semicircles of centres number of length e, and there cannot be }(25-1):, +}(25-1)", supposed of the same radius, are at once w=ptio such that we p<(n+1)e, veo<(+1)e, where , v are 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 for which both p and a 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.zf." ** tan *dx. The contri period w, and in particular those periods iw wherein o<€1, there is à 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 I* **T is parts both lie between -g and & a least value of , say lo- If tan iy tan iy Q = now and 1 = M1 +4', where M is an integer and one, any idy period Aw is of the form M9+'w; since, however, 2. Ma and w artiy - 7iy) are periods, so also is t'w, and hence, by the construction of no, where i tan iy, --sexp (y) exp(-y)]/[exp (y) +exp(-y)}, is a real we have X'=o, 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 MA, where M is an integer, and I a period. bution in question is numerically less than If beside w the functions have a period w' which is not a rea! $.***** dy that is than 20 multiple of w, consider all existing periods of the form ww trw' wherein #, v are real, and of these those for which ou 1, o<ver; Mtaz tan Ids, SH 1.-e tan : tan * dx.... |