variables xe, yo, where zo=xo+iyo, over the region; it will appear that F (2) is also continuous and in fact also a differentiable function of za. Supposing to be retained the same for all points zo of the region, and to be the upper limit of the possible values of e for the point to, 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 oo 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, 20), above, is satisfied for all points 2, within or upon the boundary of this sub-region, for an appropriate position of zo, within or upon the boundary of this sub-region. This is proved above as result (B). Hence it can be proved that, for a differentiable function f(z), the integral, f(s)ds has the same value by whatever path within the region we pass from 2 to 2. This we prove by showing that when taken round a closed path in the region the integral ff(z)dz vanishes. Consider first a triangle over which the condition (2, 2) holds, for some position of and every position of 2, within or upon the boundary of the triangle. Then as we have f(x)=f(20)+(3—20)F(2)+n0(x−20), where|0|<1, [f(z)dz=[f(z)-20F (20)]/dz+F(zo)/zdz+n/0(3−zo)dz, which, as the path is closed, is nfe (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 nap, where a is the greatest side of the triangle and p is its perimeter; if A be the area of the triangle, we have Ajab sin C> (a/r)ba, where is the least angle of the triangle, and hence a(a+b+c) <2a (b+c) <4; the integral ff(z)dz round the perimeter of the triangle is thus <4A/a. Now consider any region made up of triangles, as before explained, in each of which the condition (s, zo) holds, as in the triangle just taken. The integral ƒƒ(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 4K/e, where K is the whole area of the region, and a is the smallest angle of the component triangles. However small be taken, such a division of the region into a finite number of component triangles has been shown possible; the integral round the perimeter of the region is thus arbitrarily small. Thus it is actually zero, which it was desired to prove. Two remarks should be added: (1) The theorem is proved only on condition that the closed path of integration belongs to the region at every point of which the conditions are satisfied. (2) The theorem, though proved only when the region consists of triangles, holds also when the boundary points of the region consist of one or more closed paths, no two of which 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 fla) at the boundary points. For consider in the original region the function f(z)/(2-2), where zo is an interior point: this satisfies the same conditions as f(z) except in the immediate neighbourhood of Taking out then from the original region a small regular praygonal region with so as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral (dzf(z) 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 2-2-7E(10), dz/(x-2)=ide, and f(z) differs arbitrarily Intle from f() if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when vanishes, equal to z=if(a). Hence f(z) = where this integral is round boundary of the original region. From this it appears that F()=lim. meet. l-20' 2-20 f(z)—f(z) = (dif(t) 2-20 (z) = f(x, y)ds, for interior points zo, z, is a differentiable function of z, 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 uo(z)+u1(2)+... to be uniformly convergent over a region, its terms being differentiable functions of z, then its sum S(z) is a differentiable function of z, whose differential (S(t)dt coefficient, given by 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 f(z) include the region bounded by two concentric circles of radii r, R, with centre at the origin, and 2 be an interior point of this region, 1, where the integrals are both counter clockwise round the two circumferences respectively; putting in the first (-2)= 20/+1, and in the second (-20)-1=-2/20n+1 n0 I we find ƒ(20) = _£ Anzo”, wherein A.=d, taken round any 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 ro, the coefficients A, for which n<o all vanish, and the function (zo) is expressed for the whole interior | <R by a power series Σ Anzo". 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 2-c; a very important result. extends to within arbitrary nearness of this on all sides, and at the (8) If the region of definition, though not including the origin, same time the product "f(z) has a finite limit when a diminishes to zero, all the coefficients An for which n<-m vanish, and we have f(20) Amão +Am+150m+1+...+A__120 ̃1+A+A1...to ∞. Such a case occurs, for instance, when f(z) = cosec z, the number m being unity. = function of z is an unclosed aggregate of points, each of which § 6. Singular Points.-The region of existence of a differentiable is an interior point of a neighbourhood consisting wholly of points of the aggregate, at every point of which the function is definite and finite and possesses a unique finite differential coefficient. Every point of the plane, not belonging to the aggregate, which is a limiting point of points of the aggregate, such, that is, that points of the aggregate lie in every neighbourhood of this, is called a singular point of the function. About every interior point zo of the region of existence the function may be represented by a power series in -20, and the series converges and represents the function over any circle centre at zo which contains no singular point in its interior. This has been that if the region of existence of the function contains all points of proved above. And it can be similarly proved, putting 2=1/5, the plane for which ||>R, then the function is representable for all such points by a power series in or; in such case we say that the region of existence of the function contains the point := ∞0. A series in has a finite limit when [2]=o; a series in a cannot remain finite for all points z for which >R; for if, for [2]=R, the sum of a power series Za" in z is in absolute value less than M, of r however great, ano. Thus the region of existence of a function we have la,<Mr", and therefore, if M remains finite for all values if it contains all finite points of the plane cannot contain the point the; such is, for instance, the case of the function exp (z) = 22" /n!. This may be regarded as a particular case of a well-known result (87), that the circumference of convergence of any power series representing the function contains at least one singular point. As an extreme case functions exist whose region of existence is circular, there being a singular point in every arc of the circumference, however small; for instance, this is the case for the functions represented for < 1 by the series E 2", where m=n2, the series 2 also round the boundary of the original region. This form shows, however, that F(s) is a continuous, finite, differentiable function of ze over the whole interior of the original region. § 5. Applications.—The previous results have manifold appli cations. (1) If an infinite series of differentiable functions of z be formly convergent along a certain path lying with the region of definition of the functions, so that S(3) = uo(2) +u1 (2)+...+ (2)+R(3), where | R.(2) |<e for all points of the path, we have SS(x)ds-Suo(2)dz+fus (z)dz + + Sum (2)dz + SR.(z)dz, wherein, in absolute value, R.(z)dz <<L, if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent. (2) If f(x, y) be definite, finite and continuous at every point of a region, and over any closed path in the regionƒƒ(x, y)dz=o, then where m=n!, and the series 2/(m+1)(m+2) where m=a", 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 s the function, the series may be rearranged in powers of -20; as 20 be a point interior to the circle of convergence of a series representing approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in 2-2 diminish to zero; when, however, a circle can be put about 20, not containing any singular point of the function, but containing points outside the circle of convergence of the original series, then the series in zo gives the value of the function for these external points. If the function be supposed to be given only for the interior of the original circle, by the original power series, the series in 2-20 converging beyond the original circle gives what is known as an | analytical continuation of the function. It appears from what has been proved that the value of the function at all points of its region of existence can be obtained from its value, supposed given by a series in one original circle, by a succession of such processes of analytical continuation. the sum of the residues at the included poles, a very important result. Any singular point of a function which is not a pole is called an essential singularity; if it be isolated the function is capable, in the neighbourhood of this point, of approaching arbitrarily near to any 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 a.(z-20)"; it thus cannot remain finite in the immediate neighbourto that so far followed here. -00 hold. hood of the point. The point is necessarily an isolated essential ... ... Starting with a convergent power series, say in powers of z, this series can be arranged in powers of 2-2, about any point zo interior to its circle of convergence, and the new series converges certainly for 12-20-120, if be the original radius of convergence. If for every position of so this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence. If for some position of 20 the derived series converges for z-20] <7-|z0|+D, then it can be shown that for points z, interior to the original circle, lying in the annulus <12-20|<r|20|+D, the value represented by the derived series agrees with that represented by the original series. If for another point a interior to the original circle the derived series converges for 1-2<r-+E, and the two circles z-zo ▼ − |≈| +D, | z − ¿r-+E have interior points common, lying beyond 27, then it can be shown that the values represented by these series at these common points agree. Either series then can be used to furnish an analytical continuation of the function as originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined by an aggregate of power series of which every one has points of convergence common with some one or more others; the whole aggregate of points of the plane which can be so reached constitutes the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have radii of convergence diminishing indefinitely to zero; these are the singular points. The circle of convergence of any of the series has is capable of a form Σ b2′′, but - Σ b′′ remains finite for -0 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 f(s) is a rational ence to the single origin, by one power series, of the expressions representing the function; it is also sometimes called a monogenic analytical function, or simply an analytical function; all that is necessary to define it is the value of the function and of all its differential coefficients, at some one point of the plane; in the method previously followed here it was necessary to suppose the function differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, that ff(z)dzo over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued-it may quite well happen that starting from a particular power series, converging over a certain circle, and applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The notion of basing the theory of functions on the theory of power series is, after Newton, largely due to Lagrange, who has some interesting remarks in this regard at the beginning of his Théorie des fonctions analytiques. He applies the idea, however, primarily to functions of a real variable for which the expression by power series is only of very limited validity; for functions of a complex variable probably the systematization of the theory owes most to Weierstrass, whose use of the word monogenic is that adopted above, In what follows we generally suppose this point of view to be regarded as fundamental. § 8. Some Elementary Properties of Single Valued Functions.A pole is a singular point of the function f(z) which is not a singularity of the function 1/f(2); this latter function is therefore, by the definition, capable of representation about this point, zo, by a series [f(z)] ̄1=Σan(z—20)". If herein a, is not zero we can hence derive a representation for f(z) as a power series about zo, contrary to the hypothesis that zo is a singular point for this function. Hence doo; suppose also a1 =0, 2 =0,... am-1=0, but amo. Then [()=(−20) TM [0m+0m+1 (2−-20)+...], and hence (-20)f(z) = am1+Σb2(2−zo)", namely, the expression of f(z) about = contains a finite number of negative powers of 2-0 and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity. The integral ff(2)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 2riA, where A, is the coefficient of (2-2) in the expansion of f(z) at the pole; this coefficient has therefore a certain uniqueness, and it is called the residue of f(2) at the pole. Considering a region in which there are no other singularities than poles, all these being interior points, the integral f(x)ds round the boundary of this region is equal to function. let f.(2) = 1 81 M-1 1 If for a single valued function F(3) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to be a1, as, as,... with all and limit ao. About a, the function is expressible as A.(s—a,)"; A"(z-a.)" be the sum of the negative powers in this expansion. Assuming 2-0 not to be a singular point, let f.(:) be expanded in powers of z, in the form E C.2", and μ, be chosen so that F.(z) =f.(2)—C"C"is, for | <r,<c., less in absolute value than the general term, of a fore-agreed convergent series of real positive terms. Then the series (z) = £ F.(2) converges uniformly in any finite region of the plane, other than at the points a and is expressible about any point by a power series, and near a., (2)-f.(2) is expressible by a power series in -a.. (2)-(z) is an integral function. In particular when all the finite of an integral function and a series of rational functions. The singularities of F(s) are poles, F(z) is hereby expressed as the sum condition F.(2) <. is imposed only to render the series F,(3) uniformly convergent; this condition may in particular cases be satisfied by a series ZG,(z) where G,(z) =f.(z) - C and v.< An example of the theorem is the function cot - for which, taking at first only half the poles, f.(2)=1/(-s); in this case the series EF,(z) where F.(2)=(-s)+s1 is uniformly convergent: thus cot rs-2-1-£ [(-s)-'+5], where so is excluded from the summation, is an integral function. It can be proved that this integral function vanishes. Considering an integral function f(z), if there be no finite positions of for which this function vanishes, the function [f()] is at once seen to be an integral function, (2), or f(z) = exp []; if however great R may be there be only a finite number of values of z for which (2) vanishes, say za...am, then it is at once seen that f(z) = exp [(z)]., (z-a1)... (2-am), where (2) is an integral function, and h....h are positive integers. If, however, f(z) vanish for z == 01 as,...where a = 0. and limit a, and if for simplicity we assume that -o is not a zero and all the zeros a, a, are of the first order, we find, by applying the preceding theorem to the function 1df(2), that f(z) = exp [ø(2)] ♬ {(1 −2/an) exp ø-(1)}, where C is a constant, and F(x) is a function expressible when x is real and positive by the integral f roo 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 ƒ=ΣA,(l− a) ̃”, function f(z), and the law of increase of the coefficients in the series and differing, outside R or upon the boundary of R, from f, f(z) = 2a," as n increases (see the bibliography below under Integral Functions). It can be shown, moreover, that an integral function in the fact that while f is infinite at t=a, F is infinite only at actually assumes every finite complex value, save, in exceptional =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 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 R 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 Ro functions of a function whose singular points have the point arbitrarily little from the former, whose poles are all at the = as their only limiting point, the other, Weierstrass's point c. factor theorem, giving the expression of an integral function as a product of factors each with only one zero in the finite part of the plane, may be respectively generalized as follows: C and the interior of R can be represented at all points z in R, by Now any monogenic function f(t) whose region of definition includes f(2) sum I 2-1 S= f(li) (li+1—4) 1. If a1, G2, G3,... 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 where the path of integration is C This integral is the limit of a of the points a, a,..., and with every point a, there be associated a polynomial in (-a), say g; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point as a pole whereat the expansion where the points f, are upon C; and the proof we have given of the consists of the terms g., together with a power series in 2-a.; existence of the limit shows that the sum S converges to f(z) unithe function is expressible as an infinite series of terms giformly in regard to 2, when z is in Re, so that we can suppose, when where y, is also a rational function. far, that the subdivision of C into intervals +1-4, has been carried sufficiently II. With a similar aggregate (a), with limiting points (c), suppose with every point a there is associated a positive integer r. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order r, at the point a, but not elsewhere, expressible in the form II (1-C) exp (8)» 41-2 IS-f(z)|<e, for all points of Ro, where e is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to Ro. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from f(z), for all points of Ro, with poles at arbitrary positions outside Ro from the points of C. where with every point a, is associated a proper point c, of (c), and which can be reached by finite continuous curves lying outside R being a properly chosen positive integer. If it should happen that the points (c) determine a path dividing the plane into separated regions, as, for instance, if a, R(1-n-1) exp (ir 2.8), when(c) consists of the points of the circle =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 R, of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called Co. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of Co; let a, b, .. be finite points upon C or outside R. Then when b is near enough to a, the fraction (a-b)/(z-b) is arbitrarily small for all positions of z; say la-b |<e, for|a-b]<n; 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 number of sides of C continually greater, we can find a rational function differing arbitrarily little from f(z) for all points of R, whose poles are at one finite point c external to г. By a transformation of the form -cr-, with the appropriate change in the rational function, we can suppose this point c to be at infinity, in which case the rational function becomes a polynomial. Suppose 1, 2,.. to be an indefinitely continued sequence of real positive numbers, Co,P-f(2) <; then the infinite series of polynomials converging to zero, and P, to be the polynomial such that, within P()+(P2(2) P1(2)}+{P2(3) − P2(2) } +.... whose sum to n terms is P,(z), converges for all finite values of s and represents f(z) within Co. When C consists of a series of disconnected polygons, some of which may include others, and, by increasing indefinitely the number of sides of the polygons C, the points C become the boundary points I of a region, we can suppose the poles of the rational function, constructed to approximate to f(z) within Ro, to be at points of r. A series of rational functions of the form H1(2)+(H2(2)-H1(2)}+{H,(c)−H2(2)}+....... then, as before, represents f(z) within Ro. And Ro may be taken to coincide as nearly as desired with the interior of the region bounded by г. § 11. Expression of (1-2)-1 by means of Polynomials. Applications. We pursue the ideas just cursorily explained in some further detail. putting the com1+c by means Let c be an arbitrary real positive quantity; plex variable =+in, enclose the points =1, of (i.) the straight lines = a, from 1 to 1+c, (ii.) a semicircle convex to o of equation (-1)+2=a2, (iii.) a semicircle concave too of equation (1-c)2+na2. The quantities and a are to remain fixed. Take a positive integer so that | is less than unity, and put =(). Now take G1 =1+c/r, c2 =1+2c/r,... G=1+c; | (1-5)~1-U]<a ̄1{(1 +0*1) (1 +0"?)"i (1 +0^3)","2. o""; then, as I +x <e, we have ma |(-3)-LU]<a1{exp (0°1+n ̧0®2+n120s+ and therefore less than and therefore less than e. constructed with an arbitrary aggregate of real positive numbers 1, 2, 3,... with zero as their limit, converges uniformly and represents (1-2)-1 for the whole region considered. § 12. Expansion of a Monogenic Function in Polynomials, over a Star Region-Now consider any monogenic function f(2) of which the origin is not a singular point; joining the origin to any singular point by a straight line, let the part of this straight line, produced beyond the singular point, lying between the singular point ands=∞, be regarded as a barrier in the plane, the portion of this straight line from the origin to the singular point being erased. Consider next any finite region of the plane, whose boundary points constitute a path of integration, in a sense previously explained, of which every point is at a finite distance greater than zero from each of the barriers before explained; we suppose this region to be such that any line joining the origin to a boundary point, when produced, does not meet the boundary again. For every point x in this region R we can then write where f(x) represents a monogenic branch of the function, in case it be not everywhere single valued, and is on the boundary of the region. Describe now another region R, lying entirely within R, and let x be restricted to be within R, or upon its boundary; then for any point on the boundary of R, the points z of the plane for which zt is real and positive and equal to or greater than 1, being points for which |2|=|| or|=|>, are without the region Re, and not infinitely near to its boundary points. Taking then an arbitrary real positive e we can determine a polynomial in xt-1, say P(xt-1), such that for all points x in R, we have |(1−xt-1)-LP(xt ̄1)|<e; the form of this polynomial may be taken the same for all points ! | 2rif(x) – Sdƒ(1) P(xt1)| = | ƒ¢ƒƒ«E |¤«LM, where L is the length of the path of integration, the boundary of R, and M is a real positive quantity such that upon this boundary •Mr_1014)-1}, | |tf(t)<M. If now P(zt)=+axt and The rational function U, with a pole at =1+c, differs therefore from (1-5), for all points outside the closed region put about =1, 1+c, by a quantity numerically less than . So long as a remains the same, and will remain the same, and a less value of e will require at most an increase of the numbers n1, ng,... n; but if a be taken smaller it may be necessary to increase 7, and with this the complexity of the function U. (c+1) 3; εξ Now put }= thereby the points =0, 1, 1+c become the points =0, 1, 0, the function (1-2)- being given by (1− 3)−1 ≈ c (c + 1) −1 (1−5)~1+(c+1)~'; the function U becomes a rational function of 2 with a pole only at z=∞, that is, it becomes a polynomial in z, say1H-2, where H is also a polynomial in 2, and this gives If(x) - {coμo+c1x+...+CH"} |«LM/2π, where the quantities, 1, 2,... are the coefficients in the expansion of f(x) about the origin. If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points of f(x) to xo, it is possible, corresponding to an arbitrary real positive number, to determine a number m, and a polynomial Q(x), of order m, such that for all interior points of this region f(x)-Q(x) | <0. Hence as before, within this region f(x) can be represented by a series of polynomials, converging uniformly; when f(x) is not a single valued function the series represents one branch of the function. The same result can be obtained without the use of Cauchy's integral. We explain briefly the character of the proof. If a monogenic function of 1; $(4) be capable of expression as a power series -x about a point x, for t-xp, and for all points of this circle <g, we know that )(x)|<go""(n!). Hence, taking [2]<p, and, for any assigned positive integer, taking m so that for n>m we have (+n)<(1)", we have R n! and therefore the points (o, = 1−a), (n=0, έ=1+c+a) become respectively y= (x + c)2 + (x + c) = {} 2 μ 2μ where u=c(c+1)/a; its ordinate y, for a given value of x, can therefore be supposed arbitrarily small by taking a sufficiently small. We have thus proved the following result; taking in the plane of z any finite region of which every interior and boundary point is at a finite distance, however short, from the points of the real axis for which 1x, we can take a quantity a, and hence, with an arbitrary c, determine a number; then corresponding to an arbitrary. we can determine a polynomial P., such that, for all points interior to the region, we have Now draw barriers as before, directed from the origin, joining the singular point of (3) to z=0, take a finite region excluding all these barriers, let o be a quantity less than the radii of convergence of all the power series developments of (2) about interior points of this region, so chosen moreover that no circle of radius p with centre at an interior point of the region includes any singular point of(z), let g be such that | (2)|<g for all circles of radius p whose centres are interior points of the region, and, x being any interior point of the region, choose the positive integer n so that|xl<p; then take the points ax/n, ar=2x/n, as=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, s respectively by o and x/n, we have where hλ++...+^n, K=λ1! da!. . . An!, m1 = 132n, M2 = p2n~2,.... M2=x2, | < | <2g/2". By this formula (x) is represented, with any required degree of accuracy, by a polynomial, within the region in question; and thence can be expressed as before by a series of polynomials converging uniformly (and absolutely) within this region. § 13. Application of Cauchy's Theorem to the Determination of Definite Integrals.-Some reference must be made to a method whereby real definite integrals may frequently be evaluated by use of the theorem of the vanishing of the integral of a function of a complex variable round a contour within which the function is single valued and non singular. We are to evaluate an integral fof(x)dx; we form a closed contour of which the portion of the real axis from x=a to x=b forms a part, and consider the integral ff(z)dz round this contour, supposing that the value of this integral can be determined along the curve forming the completion of the contour. The contour being supposed such that, within it, f(z) is a single valued and finite function of the complex variable z save at a finite number of isolated interior points, the contour integral is equal to the sum of the values of ff(z)dz taken round these points. Two instances will suffice to explain the method. (1) The integral tandx is convergent if it be understood to mean the limit when e, §, o, . . . all vanish of the sum of the integrals -e tan x roo tan tan z Finally, for the remaining part of the contour, for which, with R=nx sec a, we have z = R(cos +i sin 0) = RE(0), we have exp(-R sin 0)E(iR cos 0)-exp(R sin @)E(iR cos), ide, i tanz= exp(-R sine)E(iR cos 0) +exp(R sin @)E(→ R cos 0)* when n and therefore R is very large, the limit of this contribution to the contour integral is thus dz wherein a is real quantity such that o<a<1, and the contour consists of a small circle, z=E(10), terminated at the points x= cos a, y= ± sin a, where a is small, of the two lines y sin a for cos axR cos 8, where R sin ẞ=r sin a, and finally of a large circle z=RE(14), terminated at the points x = R cos ẞ, y = R sin 8. We suppose a and ẞ both zero, and that the phase of z is zero for TCOS axR cos B, y=r sin a=R sin B. Then on cos axR cos 8, y= sin a, the phase of s will be 2, and 2-1 will be equal to exp [2ri(a-1)], where x is real and positive. The two straight portions of the contour will thus together give a contribution [1-exp (2ria)] R cos 8 x8-1 dx. It can easily be shown that if the limit of zf(z) for zo is zero, the integral ff(s)ds taken round an arc, of given angle, of a small circle enclosing the origin is ultimately zero when the radius of the circle diminishes to zero, and if the limit of zf(3) for z=∞ is zero, the same integral taken round an arc, of given angle, of a large circle whose centre is the origin is ultimately zero when the radius of the circle increases indefinitely; in our case with f()=8-1/(1+2), we have 2f(z) =2°/(1+2), which, for o<a <1, diminishes to zero both for z=0 and for so. Thus, finally the limit of the contour integral when 7=0, R=∞ is 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. jate x 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=-n to x=+n, 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 points *=*, *=*,..., the contour containing also the lines such that for every points of the interior of the region of existence z= and x-n for values of y between o and n tan a, where a of f(z) we have f(z+w)=f(z). This involves, considering all existing is a small fixed angle, the contour being completed by the portion periods w=p+io, that there exists a lower limit of p2+ other than of a semicircle of radius nr sec a which lies in the upper half of the zero; for otherwise all the differential coefficients of f(z) would be plane and is terminated at the points x= =nx, y=n tan a. Round zero, and f(2) a constant; we can then suppose that not both p and are numerically less than e, where >c. Hence, if g be any this contour the integral dz has the value zero. The contri- real quantity, since the range (-g,; g) contains only a finite number of intervals of length e, and there cannot be two periods butions to this contour integral arising from the semicircles of centres w=ptio such that μep(μ+1)e, ve=o<(+1)e, where μ, are (25-1), +(25-1), supposed of the same radius, are at once seen to have a sum which ultimately vanishes when the radius of the for which both p and are in the interval (-g...g). Considering integers, it follows that there is only a finite number of periods semicircles diminishes to zero. The part of the contour lying on then all the periods of the function which are real multiples of one rom tan Ads. the real axis gives what is meant by.2 The contri-period w, and in particular those periods Aw wherein o<1, there is a lower limit for λ, 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=27 is Sww tan a idy tan iy \nxtiy x x tan iy -nr+iy) parts both lie between -g and g, a least value of A, say X- If 2=Aow and A-MA+A', where M is an integer and o<, any period Aw is of the form M2+X'w; since, however, 2. Ma and Aw are periods, so also is X'w, and hence, by the construction of λo, we have Xo, thus all periods which are real multiples of ware where i tan iy, -[exp (y)-exp (−y)]/[exp (y)+exp (y)), is a real 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 |