+ as before there is a least value for », actually occurring in one or if a denote the generalized logarithm, -w{ALS(20+w'))-11/(2.)]], that more periods, say in the period. Y=yow twow'now take, if uw trw' is, since f(20+w) = (20), gives ?riNw, where N is an integer ; similarly be a period, r=N'nt. where N is an integer, and or'<vo; the result of the integration along the other two opposite sides is of thence uw trw'ww + N'(s' How to'w'; take then u-N'wo - Ndo+X', the form 2riN'w', where N' is an integer. The integral, however, where N is an integer and to is as above, and ozi'<do. we is equal to zri times the sum of the residues of f'(a) f(2) at the poles thus have a period NA+N'S+X'w tu'w', and hence a period interior to the parallelogram. For a zero, of order m, of (:) at =a, i'w tv'w', wherein X <te, '<n; hence m=0 and X=o. All the contribution to this sum is 2tima, for a pole of order n at := b periods of the form ww trws' are thus expressible in the form the contribution is -2 rinb; we thus infer that Emo-Enb=Nw+N'w'; NO+N'S', where a dare periods and N, N' are integers. But this we express in words by saying that the sum of the values of 3 in fact any complex quantity, P+iQ, and in particular any other where f(x) = o within any parallelogram is equal to the sum of the possible period of the function, is expressible, with u, v real, in the values of 2 where f(1) = = 0 save for integral multiples of the periods. form uw trw': for if weptio, w=p' tio', this requires only | By considering similarly the function f(2)-A where A is an arbitrary Paretrp'. Q = ro tro'. equations which, since w/w is not real, constant, we prove that each of these sums is equal to the sum of always give finite values for u and v. the values of 2 where the function takes the value A in the paralIt thus appears that if a single valued monogenic function of 2 lelogram. be periodic, either all its periods are real multiples of one of them, We pass now to the construction of a function having two and then all are of the form Ma, where N is a period and M is an integer, or else, if the function have two periods whose ratio is not arbitrary periods w, w of unreal ratio, which has a single pole real, then all its periods are expressible in the form NO+N's', of the second order in any one of its parallelograms. where 12, stare periods, and N, N'are integers. In the former case, For this consider first the network of parallelograms whose corners putting s=27120, and the function (2) =0(5), the function 018 are the points N=mw+m'w", where m, m' take all positive and has, like exp (5), the period 2ri, and if we take ? = exp (5) or <= a(l) negative integer values; putting a small circle about each corner the function is a single valued function of l. If then in particular ;(2) of this network, let P be a point outside all these circles; this will is an integral function, regarded as a function of I, it has singularities be interior to a parallelogram whose corners in order may be denoted only for t=oand 1=00, and may be expanded in the form Onlm. by so, 29+w, 20 twtw', 20+w'; we shall denote zo, sotw by Ao, Bei Taking the case when the single valued monogenic function has this parallelogram so is surrounded by eight other parallelograms two periods w, w whose ratio is not real, we can form a network ļorming with Ilo a larger parallelogram is, of which one side, for of parallelograms covering the plane of whose angular points are instance, contains the points zo-w-w', 2-w', 20-w' +w, 20-w' +20, the points c+mw + m'w', wherein c is some constant and m, m' are which we shall denote by A, B, C, D. This parallelogram in is all possible positive and negative integers; choosing arbitrarily surrounded by sixteen of the original parallelograms, forming with one of these parallelograms, and calling it the primary parallelogram. 1, a still larger parallelogram I, of which one side, for instance, all the values of which the function is at all capable occur for points contains the points 2.-20.-26", 2-w=2w", 20-2W, 2+w->w" of this primary parallelogram, any point, ?, of the plane being, 2+2w 2w": 20+3w2w', which we shall denote by As, B., C3, Day Now consider the sum of the inverse cubes of as it is called, congruent to a definite point, 2, of the primary parallelo | E, F. And so on. gram, 2-2 being of the form mw+m'w', where m, m' are integers. the distances of the point p from the corners of all the original Such a function cannot be an integral function, since then, if, in the parallelograms. The sum will contain the terms primary parallelogram \{(z)<M, it would also be the case, on a circle of centre the origin and radius R, that \|(z)<M, and therefore, if PB Eames be the expansion of the function, which is valid for an integral and three other sets of terms, each infinite in number, formed in a function for all finite values of s, we should have lanl<MR, which similar way. can be made arbitrarily small by taking R large enough. The A,B,C, A,B,C,D,E,, and so on, be p. p+a, p+29 and so on, the If the perpendiculars from P to the sides A.Be. function must then have singularities for finite values of s. sum So is at most equal to 3 2n+1 since then those of these poles which are sufficiently near to one of the necessarily existing limiting points of the poles would be of which the general term is ultimately, when n is large, in a ratio of arbitrarily near to one another, contrary to the character of a pole: equality with 29-077, so that the series So is convergent, as we know grams so Chosen that no pole falls on the perimeter of a parallelogram, the proof for the convergence of so-1/PA; is the same. Taking it is clear that the integral za Sr(a)dz round the perimeter of the the three other sums analogous to So we thus reach the result that the series primary parallelogram vanishes; for the elements of the integral (2) =2E(2-1)-9, corresponding to two such opposite perimeter (points as 2, stw (or as 2, 3+w) are mutually destructive. This integral is, however, where N is mw+m'w, and m, m' are to take all positive and negative equal to the sum of the residues of f(x) at the poles interior to the integer values, and z is any point outside small circles described with parallelogram. Which sum is therefore zero. There cannot there- the points nas centres, is absolutely convergent. Its sum is therefore fore be such a function having only one pole of the first order in independent of the order of its terms. By the nature of the proof, any parallelogram; we shall see that there can be such a function which holds for all positions of 2 outside the small circles spoken of, with two poles only in any parallelogram, each of the first order, the series is also clearly uniformly convergent outside these circles. with residues whose sum is zero, and that there can be such a function Each term of the series being a monogenic function of the series may with one pole of the second order, having an expansion near this pole therefore be differentiated and integrated outside these circles, and of the form (2-a)-?+(power series in 3-2). represents a monogenic function. It is clearly periodic with the Considering next the function $(3) ={(2)1-01043), it is easily seen periods w, w'; for (2+w) is the same sum as o(a) with the terms in a slightly different order. Thus •(3+w) = (3) and (3+w')= o(s). Consider now the function that an ordinary point of f(z) is an ordinary point of $(3), that a zero of order m for f(z) in the neighbourhood of which {(2) has a form, s(a) =$+${(2) +3}ds, (s-a)" multiplied by a power series, is a pole of 0(z) of residue m, and that a pole of f(z) of order n is a pole of o(s) of residue mi where, for the subject of integration, the area of uniform convergence manifestly $(3) has the two periods of f(2). We thus infer, since the clearly includes the point z=0; this gives sum of the residues of $(2) is zero, that for the function s(2), the sum of the orders of its vanishing at points belonging to one parallelogram, Em, is equal to the sum of the orders of its poles, En; which is and briefly expressed by saying that the number of its zeros is cqual to 5(e)-$+x{2-0-8} the number of its poles. Applying this theorem to the function (2)-A, where A is an arbitrary constant, we have the result, that wherein E' is a sum excluding the teriu for which m=0 and me' =o. the function f(3) assumes the value A in one of the parallelograms Hence (8+w)-1(z) and f(a+wisje) are both independent of . as many times as it becomes infinite. Thus, by what is proved above, Noticing, however, that, by its form, f(s) is an even function of s, every conceivable complex value does arise as a value for the doubly and putting :=-1W, :=-*W respectively, we inser that also [(s) periodic function f(e) in any one of its parallelograms, and in fact has the two periods w and w'. In the primary parallelogram Do, at least twice.' The number of times it arises is called the order of the however, f(s) is only infinite at :=o in the neighbourhood of which function; the result suggests a property of rational functions. its expansion is of the form 5'+ (power series in z). Thus {() is in round the perimeter of the primary parallelogram; the contribution It can be shown that any single valued meromorphic function to this arising from two opposite perimeter points such as sand stw of 2 with w and w'as periods can be expressed rationally in terms is of the form-w wSreda, which, as z increases from zo to zo+w'r gives, where A, B are constants. of /(?) and (s), and that ($(z) is of the form 4(3)Þ+AY(s) +B, +6+nget... Consider further the integrals - Meds, where f*(z) =449, taken any parallelogram of periods only one pole, of the second' order. 23 $(s)=(-A) (5-A:) To prove the last of these results, we write, for 131 <101, functions f(2+1), 4(:+1), which are such doubly periodic function of s as have been discussed, can each be expressed, so far as they depepd (----- stus +.... on 2, rationally in terms of S(:) and (2), and therefore, so far as they and hence, if E's-= on, since Z'820-1)=0, we have, for sufficiently | It can in fact be shown, by reasoning analogous to that given above, depend on : and I, rationally in terms of f(t), f(1), () and (4). small z greater than zero, (2) = * +307.2+507.84+.. that and j(s+1 +f(2) +f(1) = (2)=-200+609.6+2003 80+...i Lf)-1) using these series we find that the function This shows that if F(E) be any single valued monogenic function F(x) =($(3)) -4()]+60015(2)+1400; which is doubly, periodic and of meromorphic character, then contains no negative powers of s, being equal to a power series in z' F.(2+1) is an algebraic function of F(z) and F(t). Conversely any beginning with a term in st, The function F(3) is, however, doubly which is such that F(3+) is an algebraic function of F(s) and F(); single valued monogenic function of meromorphic character, F(2), periodic, with periods w, w, and can only be infinite when either f(s) or (2) is infinite; this follows from its form in f(s) and o(); can be shown to be a doubly periodic function, or a function obtained from such by degeneration (in virtue of special relations connecting thus in one parallelogram of periods it can be infinite only when the fundamental constants). s=0; we have proved, however, that it is not infinite, but, on the The functions f(2), o(s) above are usually denoted by F(s), '(); contrary, vanishes, when z=0. Being, therefore, never infinite for finite values of 2 it is a constant, and therefore necessarily always further the fundamental differential equation is usually written zero. Putting therefore f(x)=5 and (z) =ds/dz we see that (P'z): = 4(P2)-82B2-85. d: and the roots of the cubic on the right are denoted by es, es, es; *5=(458–60025–14003)-4. for the odd function, B'2, we have, for the congruent arguments Historically it was in the discussion of integrals such as -w and fw, B'($w)= -$'{-w)= - B'(tw), and hence Pfw) = 0; hence we can take eq=P(w), ez = B(lwrefw'), 6:=P(fw'). It can $85(453-6009.5-14003), then be proved that (P(c)-e][P(3 $w-e1 = ( een) (6-es), with regarded as a branch of Integral Calculus, that the doubly periodic similar equations for the other half periods. Consider more particu. functions arose. As in the familiar case larly the function P(2)-er; like (z) it has a pole of the second order at 2=0, its expansion in its neighbourhood being of the form -f5(1-2y+d. 7?(1-0,5 +Azi+...), having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (w.w'). In fact where >=sin s, it has proved finally to be simpler to regard $ as a near its zero fw its expansion is (r-\w)P'(sw)+1(-w)P' (sw)+. function of .. We shall come to the other point of view below, ; we have seen that B'(w)=0; thus it has a zero of the second under $ 20, Elliplic Integrals. order wherever it vanishes. Thus it appears that the square root To prove that any doubly periodic function F(z) with periods [P(z) –eid!! if we attach a definite sign to it for some particular value w, w', having poles at the points z=21, ...=0m of a parallelo- of 2, is a single valued function of 2; for it can at most have two gram, these being, for simplicity of explanation, supposed to be values, and the only small circuits in the plane which could lead all of the first order, is rationally expressible in terms of 0() zero, neither of which, as we have seen, has this effect; the function to an interchange of these values are those about either a pole or a. and f(s), and we proceed as follows: is therefore single valued for any circuit. Denoting the function, Consider the expression for a moment, by fi(7), we have fil:+w) = *(), fi(8+w)= S(:); (3.1). + (5.1)n-1 it can be seen by considerations of continuity that the right sign . (5-Am in either of these equations does not vary with z; not both these where A.=f(4.), 5 is an abbreviation for f(+) and for $(2), and signs can be positive, since the function has only one pole, of the first (3.1)., (5,1).-3, denote integral polynomials in's. of respective orders order, in a parallelogram (w, w'), from the expansion of fi(8) about m and m-2, so that there are am unspecified, homogeneously function, and hence fil - "W") = -f(w'), which is not zero since 2=0, namely '(1–0,22 + ...), it follows that fi(z) is an odd entering, constants in the numerator. It is supposed that no one of the points as, ...On is one of the points mw+mw where f(3) =-; (+'w)=-(2) would then give 7.(2+w+w')=f(z). and hence If (*w')}* = 63-61, so that we have fiz+w')= - Sı(z); an equation The function (3) is a monogenic sunction of a with the periods w, wifiqw+w')=f(-4w-$w’), of which the latter is - Si(tw+')this becoming infinite (and having singularities) only when (1) $ = c or (2) one of the factors S-A, is zero. In a period parallelogram infer that filztw)=f(:), 41(2+w')=-fi(8), (+w+w')=-fi(3); would give fillw+w')= 0, while (filtwtfw')}=eel. We thus including 2=0 the first arises only for e=0; since for 'S =-0,9 is in a finite ratio to 13/2; the function $(2) for $= is not infinite The function fils) is thus doubly periodic with the periods w and provided the coefficient of juin (5,1). is not zero; thus p($) is poles at :=0, 2=w each of the first order, and zeros of the first 2w'; in a parallelogram of which two sides are w and 2w it has regular about :=0. When S-A, =0, that is f(z) = f(a.), we have :=#8, +mw tm'w', and no other values of s, m and m' being of the second order with two different poles of the first order in its order at <= $w, ? = {w+w'; it is thus a doubly periodic function We may similarly consider the functions taken that the numerator vanished to the first order in each of the parallelogram (w, 2w'). es points --. -23, ... -Gmi that is, if pla.)=B., and therefore 32(z) = 11(c)-e)!, f.(z) = {P(2)-e3]4; they give (-0.) =-B.. so that we have the m relations Jal:+w+w')=fz(z), fa(z+w)= -57(3). f:(:+w')= -f(2), fi(2+w')= $33, f. (3+w)= - f:(z), 7(2+w+w')=-fi(z). (A.,1).-B.(A.,1).m-=0; Taking u =z(e-es), with a definite determination of the constant then the function (s) will only have the m poles Q1, ...Com De- (e,-es), it is usual, taking the preliminary signs so that for s=0 noting further the m zeros of F() by ci', ...am, putting fe.') = A., each of zfı(2), 2f1(3), afa(z) is equal to +1, to put (a.)=B.', suppose the coefficients of the numerator of *() to f:(z) satisfy the further m-1 conditions sn(u) = (A,,13m+B.'(A.',1)m-:=0 Ta(z) 138) k? =(eres)/(0-es), K=fwe-es)}, iK'= w'er-e:)); for s=1, 2, ... (m-1). The ratios of the 2m coefficients in the aumerator of (2) can always be chosen so that the m+(m-1) lincar with the periods 4K, 2iK, having poles of the first order at u=ik', thus sn(u) is an odd doubly periodic function of the second order conditions are all satisfied. Consider then the ratio u=2K+iK', and zeros of the first order at u=0, u = 2K; similarly F(z)/(:): cn(u), dn (u)are even doubly periodic functions whose periods can be it is a doubly periodic function with no singularity other than the written down, and sno(34) +cn?(u) = 1, k?sn?(u) + dn (11) = 1; if x= one pole G? It is therefore a constant, the numerator of $(3) sn(u) we at once find, from the relations given here, that vanishing spontaneously in am'. We have du F(2) = AP(:), =[(1-x2)(1-kox?))-1; where A is a constant; by which F(z) is expressed rationally in if we put x=sin $ we have terms of f(x) and (f), as was desired. du When :=o is a pole of F(), say of order 1, the other poles, each of the first order, being C, ... Gm, similar reasoning can be applied to do =(1-k’sin?01-2, a function and if we call the amplitude of u, we may write o=am(u), x=sin. (5.1) A+7(5,1). am(u), which explains the origin of the notation sn(u). Similarly T5-A,). (-Am cn(:4) is an abbreviation of cos, am(u), and dn(u) of A am(u), where where k, k a. such that the greater of 2h-2m, 2k+3-2m is equal Alo) meant (1-k* sin? °)The addition equation for each of the to ?; the case where some of the poles a, ... on are multiple is functions fi(2), fz(2). fo(e) is very simple, being to be met by introducing corresponding multiple factors in the denominator and taking a corresponding numerator, f(x)+F(1) f(z)f'(1) - f(0f'(2), We give a (=log solution of the general problem below, of a different form. fz) --() f'(s) -- f'(1) One important application of the result is the theorem that the I where fi'(:) means dfı(z)/ds, which is equal to - Sa(s).fo(2), and f*(3) cn(u) fi(s) = 17.(w) – B. (8) means [f(z))? This may be verified directly by showing, if R denote periods, we obtain, since the sum of the residues A is zero, a doubly the right side of the equation, that oRjazaaR/at; this will require periodic function without poles, that is, a constant; this gives the the use of the differential equation expression of F(a) referred to. The indefinite integral FF (c)ds can Viloja=U;()+er-ca][?(s) +er-el. then be expressed in terms of 2, functions B(8-2) and their differential and in fact we find coefficients, functions $(s-a) and functions log o(2-6). az § 15. Potential Functions. Conformal Representation in lazz General.-Consider a circle of radius c lying within the region hence it will follow that R is a function of 3+1, and R is at once seen of existence of a single valued monogenic function, stis, of to reduce to f(s) when 1=0. From this the addition, equation for the complex variable z,=x+iy, the origin :=o being the centre if su, c, d, s, Ca, d, denote respectively sn(u), cn(x), dn(u), sn(uz); of this circle. If z=E(io)=r(cos ¢+i sino) be an internal point cn(uz), dn(uz), they can be put into the forms of this circle we have #tive dt. where U+iV is the value of the function at a point of the cirwhere Drink's s. The introduction of the function fi(2) is equivalent to the intro. cumference and 1=0E(id); this is the same as duction of the function P(z; w, 2w) constructed from the periods utiv-S ( I (U+IV)[1-(1/a)E(10 – i )) do. w, 2w' as was P(2) from w and w'; denoting this function by B.(2) and its differential coefficient by Bi(3), we have in fact If in the above formula we replace z by the external point Bi(3) (a+/-)E(io) the corresponding contour integral will vanish, so that also as we see at once by considering the zeros and poles and the limit of sfi(3) when z=0. In terms of the function F.(2) the original function 1+(/a)-27a) cos (0-0) ) .do; B(3) is expressed by hence by subtraction we have B(s) - P.(z)+B.(s tw')-B.(w'), as a consideration of the poles and expansion near s=0 will show. U(a?-1) do, A function having w,w' for periods, with poles at two arbitrary points a, b and zeros at a', b', where a'+b=a+b save for an expres, and a corresponding formula for o in terms of V If O be the sion mw tm'w', in which m, m' are integers, is a constant multiple of centre of the circie, & be the interior point 3, P the point a E(10) (Bl:-|(a'+b')]-Pla'- }(a'+b')]} / {P12-1(a+b)}-Bla-f(@+b)]} ; of the circumference, and w the angle which QP makes with OQ if the expansion of this function near s= a be produced, this integral is at once found to be the same as. 1(3-0)4+r+E(3-0)". Sud-Sudo the expansion near :=b is -1(3-6)-+*+ E (-1)**-(8-6)*, of which the second part does not depend upon the position of s, as we see by remarking that it 2-0=-(3-0) the function has the and the equivalence of the integrals holds for every arc of same value at sands; hence the differential equation satisfied integration. by the function is easily calculated in terms of the coefficients in Conversely, let Ự be any continuous real function on the circumthe expansions. ference, U, being the value of it at a point Po of the circumference, From the function P(z) we can obtain another funetion, termed the and describe a small circle with centre at Po cutting the given circle in Zeta-function; it is usually denoted by $(z), and defined by A and B, so that for all points P of the arc AP B we have U-Uel< where e is a given small real quantity. Describe a further circle, (2){ d2 = ' centre P, within the former, cutting the given circle in A and B', and let O be restricted to lie in the small space bounded by the arc for which as before we have equations A'P.B' and this second circle; then for all positions of P upon the $(3+w) = $(2) + 2min. $(2+w')= $(x) +2 Tin', greater arc AB of the original circle QP: is greater than a definite where an, 27" are certain constants, which in this case do not both | ħnite quantity which is not zero, say QP?> ? Consider now the vanish, since else $(z) would be a doubly periodic function with only integral one pole of the first order. By considering the integral (Q'-1) i 55(z)d: 0-0)d0 round the perimeter of a parallelogram of sides w, we containing which we evaluate as the sum of two, respectively along the small arc 2=0 in its interior, we find yw'n'w=1, so that neither of n. AP.B and the greater arc AB. It is easy to verify that, for the is zero. We have 5'(z) =-P(z). From $(+) by means of the equation whole circumierence, do = we determine an integral function o(3), termed the Sigma-function, Hence we can write having a zero of the first order at each of the points := 2; it can be seen to satisfy the equations o(z+w) ol:+w'). 0(2) SEU-v.cepe By means of these cquations, if antast... tom=c' Pe'st .. If the finite angle between QA and QB be called and the finite ta'm, it is readily shown that angle AOB be calle e, the sum of the first two components is ol2-a')o(z-a's)...012-0'm) numerically less than 0(2-01)0(2-0)...013-am) (4+0). is a doubly periodic function having 41.. m as its simple poles: If the grcatest value of I (U-U.)! on the greater arc AB be called H. and a'l,...a'm as its simple zeros. Thus the function o(2) has the important property of enabling us to write any meromorphic doubly the last component is numerically less than periodic function as a product of factors cach having one zero in the Di(e-p), parallelogram of periods; these form a generalization of the simple factors, 2-6, which have the same utility for rational functions of z. of which, when the circle, of centre Pp. passing through A'B' is We have $(3) = 0' ()o(2), sufficiently small, the factor 02_2 is arbitrarily small. Thus it The functions $(2), B(3) may be used to write any meromorphic appears that u' is a function of the position of Q whose limit, when Q. doubly periodic function F(x) as a sum of terms having each only one interior to the original circle, approaches indefinitely near to Po, is pole; for is in the expansion of F(2) near a pole z=a the terms with Ve From the form negative powers of 2-a be -Sudw-psudo . A1(2-a)-:+A(2-a)*+...+Aari(3-2)(2+1), then the difference since the inclination of QP to a fixed direction is, when Q varies, P F(3) -A1$(3-0)- A, Plz - a)- + Amt'(-1)=P(m-12-0) remaining fixed, a solution of the differential cqu m! 221, , ᎾᎲ +=0, will not be infinite at 2=c. Adding to this a sum of further terms of the same form, one for cach of ihe poles in a parallelogram of where s,=x+iy, is the point Q, we inser that a' is a differentiable 00) = exp{S [s«)-1].:}=[() exp(á+)]. w-V.- Sarob (U-U.)dw Safab (U-U.)do + -Sv [1+z.cos(0-6)+acos 2(0-6)+...]d function satisfying this equation; indeed, when <a, we can write proposed for ABC; we can then determine a function for the interior (ai-) or CFAB with the boundary values so prescribed. This in its turn do will give values for the path AEC, so that we can determine a new function for the interior of AECD. With the values which this assumes along CFA we can then again determine a new function for the interior of CFAB. And so on. It can be shown that these Un tajr+bytas(?- y") +2bqxyt... functions, so alternately determined, have a limit representing where such a potential function as is desired for the interior of the original sino region ABCD. There cannot be two functions with the given de perimeter values, since their difference would be a monogenic cos 20 sin 20 potential function with boundary value zero, which can easily be de. shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem. In this scries the terms of order n are sums, with real coefficients, A particular case of the problem is that of the conformal repreof the various integral polynomials of dimension n which satisfy sentation of the interior of a closed polygon upon the upper half the equation aylar: +auloy®:,, the series is thus the real part of of the plane of a complex variablet. It can be shown without much a power series in s, and is capable of differentiation and integration difficulty that if a, b, c,... be real values of I, and a. B. 7o... ben within its region of convergence. real numbers, whose sum is n-2, the integral Conversely we may suppose a function, P, defined for the interior of a finite region R of the plane of the real variables x, y, capable as I describes the real axis, describes in the plane of z a polygon of n 2=5(1-2)--(1-6)8-1...de, series in x-xo, y-yo, with real coefficients, these various series being sides with internal angles equal to ar, Br,..., and, a proper sign obtainable from one of them by continuation. For any region Rol being given to the integral, points of the Herein the points a, b,... half of the plane of 1 interior to the region specified, the radii of convergence of these give rise to interior points of the polygon. power series will then have a lower limit greater than zero, and of the real axis give rise to the corners of the polygon; the condition hence a finite number of these power series suffice to specify the Ea=*-2 ensures merely that the point i = 0 does not correspond function for all points interior to Ro. Each of these series, and to a corner; if this condition be not regarded, an additional corner therefore the function, will be differentiable; suppose that at all and side is introduced in the polygon. Conversely it can be shown Doints of Re the function satisfies the equation that the conformal representation of a polygon upon the hall plane a P, ap can be effected in this way; for a polygon of given position of more on täys O, than three sides it is necessary for this to determine the positions of all but three of a, b, c, ; three of them may always be supposed He then call it a monogenic potential function. From this, save to be at arbitrary positions, such as t=0, 1=1,1=00. for an additive constant, there is defined another potential function As an illustration consider in the plane of 2; = x+iy, the portion by means of the equation of the imaginary axis from the origin to :=ih, where k is positive ap ap and less than unity; let C be this point z=ih; let BA be of length ar unity along the positive real axis, B being the origin and A the The functions P, Q, being given by a finite number of power series, point := ! let De be of length unity along the negative real axis, will be single valued in Ro, and P+iQ will be a monogenic function of D being also the origin and Ę the point z=-1; let EFA be a 3 within Ro. In drawing this inference it is supposed that the region semicircle of radius unity, F being the point z=i. If we put Ro is such that every closed path drawn in it is capable of being s={(z?+h?)/(1+k+zo)], with s=1 when := 1, the function is single deformed continuously to a point lying within Re, that is, is simply valued within the semicircle, in the plane of 2, which is slit along the connected. imaginary axis from the origin to :=ih; if we plot the value of 5 Suppose in particular, c being any point interior to Ro, that Pupon another plane, as a describes the continuous curve ABCDE, approaches continuously, as : approaches to the boundary of R, I will describe the real axis from s=1 to $=-1, the point C giving to the value log , where r is the distance of c to the points of the 5 =0, and the points B, D giving the points s=h. Near :=0 perimeter of R. Then the function of a expressed by $=(:-() exp(-P-IQ) the expansion of 5 is 5-h=zz! 2h +...i 2k will be developable by a power series in (-20) about every point zo in either case an increase of }in the phase of : gives an increase interior to Re, and will vanish at :=r; while on the boundary of R of a in the phase of s-hor sth. Near z=ik the expansion of 5 is it will be of constant modulus unity: Thus if it be plotted upon a 3=(:-ih)*(21h/(1k)}}+..., and an increase of 27 in the phase of plane of $ the boundary of R will become a circle of radius unity :-ih also leads to an increase of - in the phase of . Then as : closed path within Ro, passing once round :=c, will lead to a closed describes the semicircle EFA, 5 also describes a semicircle of radius path passing once about <=0. Thus every point of the interior of unity, the point 2-3 becoming $=i. There is thus a conformal R will give rise to one point of the interior of the circle. The con representation of the interior of the slit semicircle in the s-plane, verse is also true, but is more difficult to prove; in fact, the differ: upon the interior of the whole semicircle in the s-plane, the function ential coefficient dj/dz does not vanish for any point interior to R. s={(32-ho)/(1 – 175°)}} This being assumed, we obtain a conformal representation of the being single valued in the latter semicircle. By means of a transinterior of the region R upon the interior of a circle, in which the formation I=(3+1)/(5-1), the semicircle in the plane of $ can arbitrary interior point col R corresponds to the centre of the circle, further be conformably represented upon the upper half of the whole and, by utilizing the arbitrary constant arising in determining the plane of t. function Q, an arbitrary point of the boundary of R corresponds to As another illustration we may take the conformal representation an arbitrary point of the circumference of the circle. of an equilateral triangle upon a half plane. Taking the elliptic There thus arises the problem of the determination of a real mono- function P(u) for which B'?(u) = 4 Po(u)-4, so that, with e=exp (fri). genic potential function, single valued and finite within a given we have c = 1, C;= e?, es = e, the half periods may be taken to be arbitrary region, with an assigned continuous value at all points d! di of the boundary of the region. When the region is circular this Vi 2(1-1)! 2(1-1) A of argument w, and B of argument w tw'=-ew, and the equi. given. When the region is bounded by the outermost portions lateral triangle whose angular points are 0, B and C, of argument w, of the circumferences of two overlapping circles, it can hence be let E, of argument }(2w+w'), and D, of argument (w+2w'), be the proved that the problem also has a solution; more generally, concentroids of these triangles respectively, and let BE, OE, AE cut sider a finite simply connected region, whose boundary we suppose OA, AB, BO in K, L, H respectively, and BD, OD, CD cut OC, BC. to consist of a single closed path in the sense previously explained, OB in F, G, H respectively; then is u=s+in be any point of the ABCD: joining A to C by two non-intersecting paths AEC, AFC interior of the triangle OEH and v = ello elf-in) be any point of the lying within the region, so that the original region may be supposed interior of the triangle OHD, the points respectively of the ten to be generated by the overlapping regions AECD, CFAB, of which triangles OEK, EKA, EAL, ELB, EBH, DHB, DBG, DGC, DCF, the common part is AECF; suppose now the problem of determining DFO are at once seen to be given by - ev, wted, we'v, w tw' teu, a single valued finite monogenic potential function for the region wtw-v, w tw'-u, w tw' tev, w'- eu, w'ten, -e'u. Further, when AECD with a given continuous boundary value can be solved, and u is real, since the term -2(u+mw+m'e'w)-2, which is the con. also the same problem for the region CFAB: then it can be shown jugate complex of -2(u +mw+m'ew)-', arises in the infinite sum that the same problem can be solved for the original area. Taking which expresses P'(u), namely as - 2(u +uw +M'w), where indeed the values assigned for the original perimeter ABCD, assume u=m-m', u' ==-m', it follows that B'(x) is real; in a similar arbitrarily values for the path AEC, continuous with one another way we prove that o'(u) is pure imaginary when u is pure imaginary, and with the values at A and C; then determine the potential function and that B'(u) = P'(es) = P(eu), as also that for v = eu. P'(o) is the for the interior of AECD; this will prescribe values for the path conjugate complex of Þ'(w). Hence it follows that the variable CFA which will be continuous at A and C with the values originally i-fi B'(u) I-hi problem is solved by the integral - Sudw-}-Sudo previously drawing the equilateral triangle whose wervices are or of argumento [R(S.2)].-.-. takes each real value once as a passes along the perimeter of the of them by this process of continuation, a fact which we cxpress by triangle ODE, being as can be shown respectively.,1,0, - 1 at 0. saying that the equation /(s,x) = 0 defines a monogenic algebraic D, H, E, and takes every complex value of imaginary part positive construct. With less accuracy we may say that an irreducible once in the interior of this triangle. This leads to algebraic equation f(5,2) = 0 determines a single monogenic function s of 2 ratis, (R-1)+de Any rational function of zand s, where f(s.2)=0, may be considered in accordance with the general theory. in the neighbourhood of any place (c,d) by substituting therein It can be deduced that 7 = f represents the triangle ODH on the ==c+P(), s=d+Q(); the result is necessarily of the form (H(1), upper half plane of 5, and 5= (1-9-134 represents similarly the where HQ) is a power series in t not vanishing for !=0 and m is an triangle OBD. integer. If this integer is positive, the function is said to vanish to order m at the place; if this integer is negative, = -w, the function § 16. Multiple valued Functions. Algebraic Functions. The is infinite to order w at the place. More generally, il A be an explanations and definitions of a monogenic function hitherto arbitrary constant, and, ncar (c,d), R(5,2) - A is of the form IH(:), given have been framed for the most part with a view to single where m is positive, we say that R(5,2) becomes m times equal to A at the place; if R(s,) is infinite of order w at the place, so also is valued functions. But starting from a power series, say in R(s.s) - A. 'It can be shown that the sum of the values of m at all 3-C, which represents a single value at all points of its circle the places, including the places 2 = 0, where R(5.2) vanishes, which of convergence, suppose that, by means of a derived series in we call the number of zeros of R(s,z) on the algebraic construct, is 8-c, where d is interior to the circle of convergence, we can finite, and equal to the sum of the values of me where R (5,2) is infinite, continue the function beyond this, and then by means of a series R(5,2)=A; "this we express by saying that a rational function and more generally equal to the sum of the values of m where derived from the first derived series we can make a further Rs,z) takes any value including the same number of times or continuation, and so on; it may well be that when, after a the algebraic construct; this number is called the order of the rational function. closed circuit, we again consider points in the first circle of convergence, the value represented may not agree with the these values being obtainable by rational elimination of s between That the total number of zeros of R (s.s) is finite is at once obvious, original value. 'One example is the case zt, for which two values (5,2)=0, R(S.2) = 0. That the number is equal to the total number exist for any value of z; another is the generalized logarithm of infinities is best deduced by means of a theorem which is also of 1(z), for which there is an infinite number of values. In such more general utility. Let R(5,2) be any rational function of s, a, cases, as before, the region of existence of the function consists which are connected by S(s.ə) -o: about any place (c,d) for which 2=c+P(1), s=d+Q(), expand the product of all points which can be reached by such continuations with power series, and the singular points, which are the limiting R(S.2) points of the point-aggregate constituting the region of existence, in powers of land pick out the coefficient of t. There is only a are those points in whose neighbourhood the radii of convergence finite number of places of this kind. The theorem is that the sum of derived series have zero for limit. In this description the of these coefficients of r' is zero. This we express by point z=c does not occupy an exceptional position, a power series in 3-c being transformed to a series in 1/2 when z is near enough to c by means of :-(=c(1-cz-')[1-(1-cz-)|-\, and The theorem holds for the case n=1, that is, for rational functions series in 1/8 to a series in 3-c, when z is near enough to c, by 3-c=1, and about' 2 = we have r1=1, and therefore dzdie of one variable z; in that case, about any finite point we have means of -(1+: in that case, then, the theorem is that in any rational function of A Az The commonest case of the occurrence of multiple valued functions +)+2= ++R, is that in which the function s satisfies an algebraic equations(5,2) = Post + pisat + Pn=0, wherein po, Pi, ... Pa are integral poly- the sum EA, of the sum of the residues at the finite poles is equal nomials in 2. Assuming (s.2) incapable of being written as a product to the coefficient of 1/2 in the expansion, in ascending powers of 1/5, of polynomials rational in s and 2, and excepting values of 2 for about 2=c;, an obvious result. In general, if for å finite place which the polynomial coefficient of sn vanishes, as also the values of the algebraic construct associated with f(s,:) = 0, whose neighbour. of s for which beside f(5,2)=0 we have also of($,2)/as=0, and also hood is given by z=c+l.s=d+Q®), there be a coefficient of in in general the point 2=0, the roots of this equation about any point R(s,z)dzdı, this will be r times the coefficient of 17 in R(5,2) or :-c are given by a power series in 3-c. About a finite points= Rid+QUt), c++), namely will be the coefficient of or in the sum of for which the equation af(s,)/as ro is satished by one or more of the the r series obtainable from Rid+Q(). c+f) by replacing ! by wl, roots s of f(s,z) = 0, then roots break up into a certain number of where w is an rth root of unity; thus the sum of the coefficients of cycles, the r roots of a cycle being given by a set of power series in in R(s,z)dz/dt for all the places which arise for x=c, and the corre a radical (3-6)', these series of the cycle being obtainable from sponding values of s, is equal to the coefficient of (3-6) in R($1,3)+ one another by replacing (2-c)'/ by ws:-()', where w, equal to R ($2,2) + ... +R(S.,), where si, ... SA are the n values of s lor a exp (2 rih/r), is one of the rth roots of unity. Putting then 3-c=ľ value of 2 near to s=c; this latter gum ER(si, z) is, however, a we may say that the r roots of a cycle are given by a single power rational function of s only. Similarly, near :=-0, for a place given series in t, an increase of 2r in the phase of t giving an increase of by th=r, s=d+Q(1), or 's-= Q(1), the coefficient of r' in R(5,2) dedi 2nr in the phase of 3-c. This single series in t, giving the values of is equal to -r times the coefficient of ! in R[d+Q(1), r), that is s belonging to one cycle in the neighbourhood of := when the phase equal to the negative coefficient of r in the sum of the r series of :-c varies through 277, is to be looked upon as defining a single Rd+Q(wl), r*). so that, as before, the sum of the coefficients of place among the aggregate of values of z and s which satisfy f(5,2) = 0; plin 5,2)de/dt at the various places which arise for z = is equal two such places may be at the same point (2=c, s=d) without to the negative coefficient of z' in the same rational function of :, coinciding, the corresponding, power series for the neighbouring R($1,2). Thus, from the corresponding theorem for rational functions points being different. Thus for an ordinary value of 2, 3 =C, thereof one variable, the general theorem now being proved is seen to are n places for which the neighbouring values of s are given by n follow. power series in 2-c; for a value of 2 for which af(3,2)/ds=0 there Apply this theorem now to the rational function of s and 2, are less than n places. Similar remarks hold for the neighbourhood of s = 0; there may be n places whose neighbourhood is given by n R(3,3) dz power series in rl or fewer, one of these being associated with a series in I, where 1=(7)/: the sum of the values of r which thus at a zero of R(5,8) near which R(s, 3) = 1mH(1), we have arise is always n. In general, then, we may say, with t of one of ds the forms (s-c), (3-0)1/1, 7!, (7-1)/", that the neighbourhood of R(5,2) di any place (c,d) for which f(c,d) =o is given by a pair of expressions where a denotes the generalized logarithmic function, that is equal ( ==c+P(i), s=d+Q(1), where P(!) is a (particular case of a) power series vanishing for i=0, and Qui) is a power series vanishing for mrit power series in t; 1 = 0, and t vanishes at (c,d), the expression :-c being replaced by similarly at a place for which R(s,x)=t-K(1); the theorem f! when c is infinite, and similarly the expression s-d by sal when d is infinite. The last case arises when we consider the finite values of 3 for which the polynomial coefficient of s vanishes. Of such a R(5,3) dz pair of expressions we may obtain a continuation by writing i = le+ thus gives &m=y, or, in words, the total number of zeros of R(s, e) λιτ +λητ’ + where r is a new variable and is not zero; on the algebraic construct is equal to the total number of its poles. in particular for an ordinary finite place this equation simply becomes The same is therefore true of the function R(s,z) - A, where A is an i = to tr. It can be shown that all the pairs of power series z=c+ arbitrary constant; thus the number in question, being equal to the P(), s=d+Q(1) which are necessary to represent all pairs of values number of poles of R(5,2)-A, is equal also to the number of times of 3, s satisfying the equation |(9,5) = 0 can be obtained from one that R(5,5) = A on the algebraic construct. 1 dR(5.2); 1 to |