as before there is a least value for », actually occurring in one or 1 denote the generalized logarithm, -w12({(zo+w'))-115 (30)]], that more periods, say in the period. ' = How troos' now take, if uw trw' is, since f(20+)=f(20), gives 2riNw, where N is an integer; similarly be a period, -=N'nty, where N is an integer, and o'<noi the result of the integration along the other two opposite sides is of thence wwww' =ww + N'(- How to'w'; take then u-N'gus = Ndo+X, the form 2riNw. where N' is an integer. The integral, however, where N is an integer and do is as above, and ošX<dei we is equal to zri times the sum of the residues of f'(x){f(s) at the poles thus have a period NO+N'S+wu'w', and hence a period interior to the parallelogram. For a zero, of order m, of f(2) at :=a, i'w to'w'wherein X <te, '<ro; hence w=0, and X'=o. All the contribution to this sum is 2nimo, for a pole of order n at 2= 6 periods of the form wwww are thus expressible in the form the contribution is -2mind; we thus infer that Ema-Enb-Nw+N'w': NA+N's, where Be Bare periods and N, N' are integers. But this we express in words by sa ying that the sum of the values of a in fact any complex quantity, P+iQ, and in particular any other where f(x)=0 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 a where 16 ) = 0 save for integral multiples of the periods. form yw trw'; for, if, w=etio, w'-e'tio', this requires only By considering similarly the function ()-A where A is an arbitrary P=uptop'. Q=votro', equations which, since wlw is not real, constant, we prove that cach of these sums is equal to the sum of always give finite values for 4 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 Me, where A 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 Na+N'X, of the second order in any one of its parallelograms. where 1, Stare periods, and N, N'are integers. In the former case, For this consider first the network of parallelograms whose corners putting s=2ris/, and the function {(z) = o(s), the function () are the points Q=mwtm'w'. where m, m' take all positive and has, like exp (5), the period 2ri, and if we take !=exp (5) or = negative integer values, putting a small circle about cach corner the function is a single valued function of t. If then in particular f() of this network, let P be a point outside all these circles; this will is an integral function, regarded as a function of 1, it has singularities be interior to a parallelogram whose corners in order may be denoted only for i=0 and 1=00, and may be expanded in the form an. by 20, 2+w, 20+w+w', zo+w'; we shall denote zo, 20 tw by An. Bei Taking the case when the single valued monogenic function has this parallelogram mo is surrounded by eight other parallelograms, wo periods w, w whose ratio is not real, we can form a network forming with ll. a larger parallelogram ., of which one side, for of parallelograms covering the plane of a whose angular points are instance, contains the points 20-w-w', 20-w': 20-w'+w, 20-w' +20, the points c+mwtm'w', wherein c is some constant and m, m' are which we shall denote by A, B, C, D. This parallelogram II, 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. 0, a still larger parallelogram ti, of which one side, for instance, all the values of which the function is at all capable occur for points contains the points 9.-20.7.20", 20-4-2w", 2-2, 8+w-w' of this primary parallelogram, any point, of the plane being. +2w2w; 20+3w2w". which we shall denote by Ag, Ba, Cz. De as it is called, congruent to a definite point, s, of the primary parallelo: E. F. And so on. Now consider the sum of the inverse cubes of gram, 8-s 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 \f(z) <M, it would also be the case, on a circle So=PÅ3+ (PÅq +PBA +P) + (p^2+PB2+...+px) +... of centre the origin and radius R, that \{() <M, and therefore, if Edqz" 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 land<MR, which similar way can be made arbitrarily small by taking R large enough. The A,B,C.. AB,C,D,Ez, and so on, be p. P+g. P+29 and so on, the If the perpendiculars from P to the sides AoBo. function must then have singularities for finite values of z. sum S, is at most equal to We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram. 5 anti since then those of these poles which are sufficiently near to one +®+ngit of the necessarily existing limiting points of the poles would be arbitrarily near to one another, contrary to the character of a pole. equality with 2q-in?, so that the series So is convergent, as we know of which the general term is ultimately, when n is large, in a ratio of 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 S1(a)ds 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 $(z) = –23(2-0)-2, corresponding to two such opposite perimeter points as 3, stw (or as 2, 3+w) are mutually destructive. This integral is, however, where his mw +m'c', and m, m' are to take all positive and negative equal to the sum of the residues of f(s) at the poles interior to the integer values, and 2 is any point outside small circles described with. parallelogram. Which sum is therefore zero. There cannot there the points as centres, is absolutely convergent. Its sum is therefore fore be such a function having only one pole of the first order in independent of the order of its terms. By the nature of the proof, any parallelogram; we shall see that there can be such a function which holds for all positions of s outside the small circles spoken of with two poles only in any parallelogram, each of the first order, the series is also clearly uniformly convergent outside these circles. with residues whose sum is zero, and that there can be such a function Each term of the series being a monogenic function of z, the series may with one pole of the second order, having an expansion near this pole therefore be differentiated and integrated outside these circles, and of the form (2-a)-7+(power series in s-a). represents a monogenic function. It is clearly periodic with the Considering next the function o(a) =($(+)4983), it is easily seen periodismely difference order) is the same sum as ob with the terms in a slightly different . w) and w) Consider now the function that an ordinary point of f(2) is an ordinary point of $(x), that a zero of order m for f(s) in the neighbourhood of which {(z) has a form. (-a)" multiplied by a power series, is a pole of $(2) of residue m, and that a pole of f(z) of order n is a pole of $(3) of 'residue n; where, for the subject of integration, the area of uniform convergence manifestly (3) has the two periods of f(d). We thus infer, since the clearly includes the point s=0; this gives sum of the residues of $(z) is zero, that for the function ((:), 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 equal to Se) -+x{1-0-8 the number of its poles. Applying this theorem to the function (3)-A, where A is an arbitrary constant, we have the result, that wherein is a sum excluding the teruu for which m=0 and m'=0. the function S(2) assumes the value A in one of the parallelograms Henge fls+w)-f(z) and 1(2+wrs are both independent of as many times as it becomes infinite. Thus, by what is proved above, Noticing, however, that, by its form, f() is an even function of :, every conceivable complex value does arise as a value for the doubly and putting 2=-W, =-w' respectively, we infer that also sia periodic function (2) in any one of its parallelograms, and in fact has the two periods w and w. In the primary parallelogram 11.. at least twice.' The number of times it arises is called the order of the however, f(x) 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 a). Thus {(2) is Consider further the integrals-freds, where f'() "01. taken any parallelogram of periods only one pole, of the second order. , in round the perimeter of the primary parallelogram; the contribution to this arising from two opposite perimeter points such as 2 and 3+w of with w and w'as periods can be expressed rationally in terms is of the form--freds , which, as e increases from se to zota', sives, of (a) and $(8), and that [0(3)¥ is of the form 4!/(@)P+Af(x)+B. B To prove the last of these results, we write, for 1.1 <10% functions f(3+1), 4(x+1), which are such doubly periodic function of s as have been discussed, can each be expressed, so far as they depend 6-6-6- ++.... on z, rationally in terms of S(3) and (z), and therefore, so far as they and hence, if E'oro, since I's: (20-1)=0, we have, for sufficiently it can in fact be shown, by reasoning analogous to that given above, depend on x and t, rationally in terms of (:), ), (2) and (i). small & greater than zero, that f(2)=30+303.04 +509.84+.. and 81 0f (2)=-28 +609.8+2065 80+...; -100) using these series we find that the function This shows that if F(2) be any single valued monogenic function F(x) = ($(z))?- 41(s)}'+600W()+14001 which is doubly, periodic and of meromorphic character, then contains no negative powers of 2, being equal to a power series in 3 Flat) is an algebraic function of F(x) and F(). Conversely any beginning with a term in . The function F(s) is, however, doubly which is such that F (3+!) is an algebraic function of F(z) and Fi), periodic, with periods w, w, and can only be infinite when either (z) or (2) is infinite; this follows from its form in (2) and (8): can be shown to be a doubly periodic function, or a function obtained thus in one parallelogram of periods it can be infinite only when from such by degeneration (in virtue of special relations connecting the fundamental constants). 2=0; we have proved, however, that it is not infinite, but, on the The functions f(2), (2) above are usually denoted by F(s), D'(*); contrary, vanishes, when :=0. Being, therefore, never infinite for finite values of : 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/ds we see that (B'z):= 4(P2) -82B2-8.. and the roots of the cubic on the right are denoted by er, ez, és; ds di = (459-60935–14001) for the odd function, B's, we have, for the congruent arguments -fw and fw, B'(tw) - B'(-w)= - B'(jw), and hence P'lfw)=0; Historically it was in the discussion of integrals such as hence we can take a = B(w), 6. = Bwrw').e;= P(fw'). It can 5d5(458-6009.5-14001), then be proved that (B(z)-ell(2+1)-ef=(er-en) (er-cs), 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(x)-e; like B(z) it has a pole of the second order at 2=0, its expansion in its neighbourhood being of the form -$$(1-19 +d5e 8*(1-212+Az*+...); having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (w. w'). In fact wbere $ = sin s, it has proved finally to be simpler to regard & as a neat its zero fw its expansion is (x-w)$'(JW) + f(-1)'(tw)+ function of s. We shall come to the other point of view below, ; we have seen that P'(fw-o; thus it has a zero of the second ander 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(s) -eill. if we attach a definite sign to it for some particular value w, w', having poles at the points z=21, ...3=0, of a paralleloof 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 $() zero, neither of which, as we have seen, has this effect; the function and (8), and we proceed as follows: is therefore single' valued for any circuit. Denoting the function, Consider the expression for a moment, by fı(), we have 13+w)= *fi(s), fili+w')=f(); p(s) (5.1).to(st). it can be seen by considerations of continuity that the right sign T-A5-A3) . (-A) in either of these equations does not vary with 2; not both these where A.-f(e.), 5 is an abbreviation for f(z) and q for $(s), and signs can be positive, since the function has only one pole, of the first mo and m-2, 60 that there are am unspecified, homogeneously function, and hence fi(- jw") = -f(w), which is not zero since (5.1)., (5.1) m-3, denote integral polynomials in's. of respective orders order, in a parallelogram (w, w'): from the expansion of fi(8) about 2=0, namely '(1 - 10,8+ ...), it follows that fi() is an odd entering, constants in the numerator. It is supposed that no one ('wp-c-4, so that we have fil: w')= -f(s); an equation The function () is a monogenic function of 2 with of which the latter is (tw+'); this becoming infinite (and having singularities) only when (1) S=C or (2), one of the factors s-A, is zero. In a period parallelogram infer that fila+w)=f(s), V(+W) = {s(3), (+w+w')=-fi(*); would give filtwt1w)=0, while fi(tw+w')}=erenWe thus including s=0 the first arises only, for 3=0; sînce for 5 = 0,7 is in the function () is thus doubly periodic with the periods w and a finite ratio to 3!2: the function () for $= is not infinite 2w; in a parallelogram of which two sides are w and 2w it has provided the coefficient of grow in (5,1). is not zero: thus p(q) is poles at s=0, $=w each of the first order, and zeros of the first regular about 3-0. When s-A=0, that is f(x) =f(a.), we have 2=+,+mwtm'w, and no other values of sm and being of the second order with two different poles of the first order in its order at 2= fw, = fw+w'; it is thus a doubly periodic function taken that the numerator vanished to the first order in each of the parallelogram (w, zw'). We may similarly consider the functions m points . -, ... Ami that is, it (e.) – B., and therefore F.() 19.43)-elt, fo(a) LP (2)-ca)they give (+w+w)=f() (+w)=-). Sel:+w')= -2(). 01-0.) --B., so that we have the m relations (2+w')=fa, fa+w)= -fs(2), file+w+w')= -f(x). (A,,1)m-B,(A,,I)omi=0; Taking u = 3(61-es), with a definite determination of the constant then the function (?) will only have the m poles or, De (9-es), it is usual, taking the preliminary signs so that for 2-0 noting further the m zeros of F) by 01, ...emi putting f(0.)-A., each of fi(), fa(3), 2f:(2) is equal to +1, to put (Q.) = B.'. suppose the coefficients of the umerator of $(xto satisfy the further m-1 conditions sn(w) – 19.70)cn(x) = 0.0 dn () (A.',1).+B.'(A,',1)==0 for s= 1, 2,... (m-1). The ratios of the 2m coefficients in the thus sn(u) is an odd doubly periodic function of the second order p=(eres)/(e-es), K-fwles-es):. K'-fwleres)t; numerator of 6 can always be chosen so that the m+(m-1) linear with the periods 4K, 2iK, having poles of the first order at urik' conditions are all satisfied. Consider then the ratio u=2K+iK', and zeros of the first order at u=0, x= 2K; similarly F(x)/(z): en(u), dn (u)are even doubly periodic functions whose periods can be it is a doubly periodic function with no singularity other than the written down, and sn (w) +eno(x) = 1, k-sno(u) +dn* (u) = 1; il *one pole om? It is therefore a constant, the numerator of +(s) sn(x) we at once find, from the relations given here, that vanishing spontaneously in Om'. We have du F(x) = A$(3), de = 1(1-x+)(1-kox?))-1; where A is a constant; by which F() is expressed rationally in if we put =sio 6 we have terms of f(x) and $(x), as was desired. When 3=o is a pole of F(3), say of order 1, the other poles, each of. te= (1-Pain°957. the first order, being Oj, ... Come similar reasoning can be applied to a function and if we call the amplitude of u, we may write o=am(u), --sin. (SDA+,(5,1). am(u), which explains the origin of the notation sn(u). Similarly 5-A,... -AT cn(u) is an abbreviation of cos. am(u), and dn(s) of A am(u), where where k, k a.e such that the greater of 2k-2m, 2k+3+2m is equal A(6) meant (1-k sin o)! The addition equation for each of the to l; the case where some of the poles 21.1 are multiple is functions fi(3),fe().fo(a) is very simple, being to be met by introducing corresponding multiple factors in the de nominator and taking a corresponding numerator. We give a f(s+h= (detail) log f(2)=f(0)_[(2)$"(!) –SUJ"), solution of the general problem below, of a different form. $ F"(z) -S(O) One important application of the result is the theorem that the'l where fi'(:) means dfi(s)[dz, which is equal to -f(s).S.(), aod (8) .O mcans [f(z))?. This may be verified directly by showing, if R denote i periods, we obtain, since the sum of the residues A is zero, a doubly the right side of the equation, that OR/dz=arjat; this will require periodic function without poles, that is, a constant; this gives the the use of the differential equation expression of F(?) referred to. The indefinite integral fF 8)da can Visoje = 01(%)+e-ea(z)+a-cal. then be expressed in terms of s, functions B(2-a) and their differential and in fact we find coefficients, functions $(2-a) and functions log o(s-a), la? as $ 15. Potential Functions. Conformal Representation in General. Consider a circle of radius a lying within the region hence it will follow that R is a function of sti, and R is at once seen of existence of a single valued monogenic function, utis, of to reduce to ()' when =0. From this the addition equation for the complex variable z,=x+iy, the origin z=o being the centre each of the functions sn(u), cn(u), dn(u) can be deduced at once; if si, cu, d., , C3, d, denote respectively sn (21), cn(us), dn(w.), sn(us) of this circle. If := E(ip)=r(cos$+i sino) be an internal point ch(113), dn(uz), they can be put into the forms of this circle we have utiv-zisuud) di, where U+iV is the value of the function at a point of the cirwhere The introduction of the function file) is equivalent to the intro- cumference and 1=cE(18); this is the same as duction of the function P(s; w, zw) constructed from the periods *tive do. w, 2w' as was Þ(2) from w and w'; denoting this function by Bi(s) 25) i+ya)? - 2(71a) cos (0-0) and its differential coefficient by B'i(), we have in fact If in the above formula we replace z by the external point fi(z) = 1 Bi(3) (@?/r)E(ip) the corresponding contour integral will vanish, so that B.(w) – B.(2) also as we see at once by considering the zeros and poles and the limit of 2f1() when s=o. In terms of the function B.(s) the original function 1+r/a)-271a) cos((-) fig)40: B(3) is expressed by bence by subtraction we have B(x)= P.(s)+B.(: tw')-B.(w'), as a consideration of the poles and expansion near 2=0 will show. -- Sa+; -2.arcos (0-2)do. A function having w,a' for periods, with poles at two arbitrary points a, b and zeros at a', ', where a'+b=: +6 save for an expres; and a corresponding formula for o in terms of V. If O be the sion mw I m'w', in which m, m' are integers, is a constant multiple of centre of the circie, a be the interior point 2, P the point nE(id) {${s-f(a'+b))–Bla'- He'+b')}} / {\[2- }(a +-6)}-Bla-f(a+b)]} of the circumference, and w the angle which QP makes with OQ if the expansion of this function near x=0 be 1(3-0)-+++Eyn(8-6)*. produced, this integral is at once found to be the same as. --the expansion near x=b is -1(-b)-+w (-1)*un(8-1)", of which the second part does not depend upon the position of 3, as we see by remarking that if :-6=-(-a) the function has the and the equivalence of the integrals holds for every arc of same value at 2 and 7; hence the differential equation satisfied integration. by the function is easily calculated in terms of the coeficients in Conversely, let U be any continuous real function on the circumthe expansions. ference, V. being the value of it at a point Po of the circumference, From the function B(+) we can obtain another function, termed the and describe a small circle with centre at P. cutsing 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 havel U-va! Se where is a given small real quantity: Describe a further circle, ()] 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 $(2+w)= $(x) + 2rin, $(3+w')=$(2) +2rin grcater arc AB of the original circle QPis greater than a definite where 27, 29' are certain constants, which in this case do not both knite quantity which is not zero, say QP2> : Consider now the vanish, since else $(s) would be a doubly periodic function with only integral one pole of the first order. By considering the integral (a15(e)d: round the perimeter of a parallelogram of sides w, w containing which we evaluate as the sum of two, respectively along the small arc :-o in its interior, we find nu'n'w=!, so that neither of 7.7 AP B and the greater arc AB. It is easy to verify that, for the is zero. We have 5'()=-P(2). From $(e) by means of the equation whole circumference, EII' - cos 0-0) ư-U = (2) QP By means of these equations, if a tast. tam=a' 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 called e, the sum of the first two components is ols-a'l)o(2-0')...03-0'.) numerically less than o(2-1)0(2-0)....(2-x) 2 (4+0). iš a doubly periodic function having a, ...Om as its simple poles, and c'..... om as its simple zeros. Thus the function (2) has the If the greatest value of 1 (U-.) l'on the greater arc AB be called H, important property of enabling us to write any meromorphic doubly the last component is numerically less than periodic function as a product of factors each having one zero in the parallelogram of periods; these form a generalization of the simple 07(e?-rj. factors, 2-a, which have the same utility for rational functions of z. of which, when the circle, of centre P. passing through A'B' is We have $(2) = 0'(?)lo(s). sufficiently small, the factory is arbitrarily small. Thus it The functions $(3), B( ) 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(*) as a sum of terms having each only one interior to the original circle, approaches indefinitely near to Pa, is pole; for is in the expansion of F() near a pole z=o the terms with V. From the form negative powers of z-a be A1(3-2)*+A(8-a)%+...+Amrils-a) ***1), then the difference since the inclination of QP to a fixed direction is, when Q varies, P F(2) -A;5(3-0)-A,B(5-0)-... + Amy'(-1).mp3cm-13-4) remaining fixed, a solution of the differential equation will not be infinite at z=a. Adding to this a sum of further terms of the same form, one for cach of the poles in a parallelogram of where s, -x+iy, is the point Q. we infer that s' is a differentiable olstw) 21 JAB JA function satisfying this equation; indeed, when rca, we can write proposed for ABC; we can then determine a function for the interior of CFAB with the boundary values so prescribed. This in its tura Guattr2ar cos (0-4) will give values for the path AEC, so that we can determine a new function for the interior of AECD. With the values which thig -Sv [1+2 cos(0-6) +cos 2(0-6)+... ] do 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 =+0,8+by+oz(*- y") +2bqxyt. functions, so alternately determined, have a limit representing where such a potential function as is desired for the interior of the original *Sudo , -, --Su cosedo, 6,- Susiness. region ABCD. There cannot be two functions with the given perimeter values, since their difference would be a monogenic potential function with boundary value zero, which can easily be 9- + Su con 2040, 6-8 min 20 do. shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem. In this series the terms of order n are sums, with real coefficients, A particular case of the problem is that of the confort.ual repreof the various integral polynomials of dimension n which satisfy sentation of the interior of a closed polygon upon the upper half the equation avläx: +oploye: 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 2, and is capable of differentiation and integration difficulty that if a, b, c,... be real values of land a. 6. 7.... 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 sa polygon of :/(-a)2-1(1-6)A-1...dt, of expression about any interior point for Yo of this region by a power sides with internal angles equal to ar, Br...., and, a proper sign, obtainable from one of them by continuation. For any region R ) being given to the integral, points of the upper half of the plane oli interior to the region specified, the radii of convergence of these give rise to interior points of the polygon. Herein the points a, b, ... 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 2a=-2 ensures merely that the point i= does not correspond Sunction 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 points of Ro the function satishes the equation that the conformal representation of a polygon upon the half plane a? PaP can be effected in this way; for a polygon of given position of more 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 we then call it a monogenic potential function. From this, save for an additive constant, there is defined another potential function to be at arbitrary positions, such as 1=0,1-1,1=0. As an illustration consider in the plane of :=x+iy, the portion by means of the equation of the imaginary axis from the origin to suik, where h is positive rii, w) and less than unity; let C be this point z=ih; let BA be of length dx 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 :-1, 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 2=-1; let EFA be a within Ro. In drawing this inference it is supposed that the region semicircle of radius unity, F being the point 3=1. If we put Ro is such that every closed path drawn in it is capable of being s-l(*+hay/(1+k%*)*; with $=1 when s=s, the function is single deformed continuously to a point lying within Ro, that is, is simply valued within the semicircle, in the plane of 2, which is slit along the connected. 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 will describe the real axis from :=1 to $=-1, the point C giving to the value log , wherer is the distance of c to the points of the $=0, and the points B, D giving the points $= *k. Near s=0 perimeter of R. Then the function of z expressed by $=(:-() exp (-P-Q) the expansion of ļ is 5-k=345**+..., or $tho-gel 2h -t... will be developable by a power series in (3-0) about every point in either case an increase of £t in the phase of gives an increase interior to Ro and will vanish at z=c; while on the boundary of R of r in the phase of 3-h or sth. Near z=ik the expansion of s is it will be of constant modulus unity. Thus if it be plotted upon a s= (-ih)'laih/(1-•)}}+..., and an increase of 27 in the phase of plane of 5 the boundary of R will become a circle of radius unity 2-ih also leads to an increase of - in the phase of s: Then as : with centre at $=0, this latter point corresponding to 2=4. A describes the semicircle EFA, 5 also describes a semicircle of radius closed path within Ro, passing once round :-c will lead to a closed unity, the point :=i becoming, s=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 2-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 dsds does not vanish for any point interior to R. z=1(32-ho)/(1-hosa)} 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 1- (5+1)/(3-1), the semicircle in the plane of scan 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 B(u) for which B'?(u) =4 Po(u)-4. so that, with exp (ni), genic potential function, single valued and finite within a given we have ej = 1, ezze, es= 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 fw' 2(-1)! es 2(13-1) B of 'ew, and the equigiven. 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+aw'), be the proved that the problem also has a solution; more generally, con- centroids 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 if u=$+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 = €160 = (-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 AÉCD, 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 -e, wteu, we'v, wtw teu, a single valued finite monogenic potential function for the region wtwo, w tw'-u, w ta' teo, w'-eu, w'ten, -eh. Further, when AECD with a given continuous boundary value can be solved, and is real, since the term - 2(x+mw+m'ew), which is the conalso the same problem for the region CFAB; then it can be shown jugate complex of -2(x+mwtm'ow)"}, arises in the infinite sum that the same problem can be solved for the original area. Taking which expresses P'la), namely, as -2(u + uw tu' ew, where indeed the values assigned for the original perimeter ABCD, assume u=m-m', u' E-m', it follows that B'(u) is real; in a similar arbitrarily values for the path AEC, continuous with one another way we prove that Þ'(u) is pure imaginary when u is pure imaginary, and with the values at A and C; then determine the potential function and that P'(x)=B'(eu) =B (eu), as also that for o=eu, B'() is the for the interior of AECD: this will prescribe values for the path conjugate complex of D'(o). Hence it follows that the variable, CFA which will be continuous at A and C with the values originally 1-FB'(u) problem is solved by the integral sudw - *S Ude previously drawing the equilateral triangle whose vertices are o: ok azamento R(s211 takes each real value once as # passes along the perimeter of the of them by this process of continuation, a fact which we express by triangle ODE, being as can be shown respectively 0 , 1.0, -1 at 0. saying that the equation (5,2) - 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) = o determines a single monogenic function s of v=tif," (P-1)-tól Any rational function of z and s, where (5.2) =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 of represents the triangle ODH on the 2=c+P%), s = d+Q(); the result is necessarily of the form 2-H(1). upper half plane of 7, and $ = (1-4-1) represents similarly the where H() is a power series in 1 noc 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, = cm. the function f 16. Multiple valued Functions. Algebraic Functions. The is infinite to order w at the place. More generally, if A bc an explanations and definitions of a monogenic function hitherto arbitrary constant, and, near (c,d), R(5.2) -A is of the form ("H(1). 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 valued functions. But starting from a power series, say in R(5,2) - A. 'It can be shown that the sum of the values of m at all at the place; if R(5,2) is infinite of order w at the place, so also is 2-C, which represents a single value at all points of its circle the places, including the places 2 =co, where R(S.2) vanishes, which of convergence, suppose that, by means of a derived series in we call the oumber of zeros of R(5,2) on the algebraic construct, is 8~0,' where t is interior to the circle of convergence, we can finite, and equal to the sum of the values of je where R (5,2) is infinite, continue the function beyond this, and then by means of a series and more generally equal to the sum of the values of m where R(3,2) = A; this we express by saying that a rational function derived from the first derived series we can make a further R8,2) 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; chis number is called the order of the closed circuit, we again consider points in the first circle of rational function. convergence, the value represented may not agree with the That the total number of zeros of R (5.2) is finite is at once obvious, these values being obtainable by rational elimination of s between original value. One example is the case , for which two values (5,2) = 0, R(5,2)=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 X(), for which there is an infinite number of values. In such more general utility. Let R75,3) be any rational function of s, s, which are connected by /(s,z) = 0; about any place (c,d) for which cases, as before, the region of existence of the function consists 2=c+P(1), s=d+Q(i), expand the product of all points which can be reached by such continuations with power series, and the singular points, which are the limiting points of the point-aggregate constituting the region of existence, in powers of 1 and pick out the coefficient of r. There is only a are those points in wbose 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 His zero. This we express by point :=60 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 2-c=c(1-c2^^)(1-(1-2)}, and a The theorem holds for the case n= 1, that is, for rational functions series in 1s to a series in z-c, when z is near enough to c, by 2-c=1, and about z=- we have rl=h, and therefore dz/d =-; of one variable s; in that case, about any finite point we have means of this of-(+). in that case, then, the theorem is that in any rational function of a, The commonest case of the occurrence of multiple valued functions (Antojit... +2-3)=) +Pa+Q+...+R, is that in which the function s satisfies an algebraic equation (5,2) = Pos" +pistit... t.pno, wherein po. Di... Do are integral poly- the sum 2A, of the sum of the residues at the finite poles is equal nomials in z. Assuming $(5,2) incapable of being written as a product to the coefficient of 1/3 in the expansion, in ascending powers of 172, of polynomials rational in s and 2, and excepting values of 3 for about =; an obvious result. In general, if for a finite place which the polynomial coefficient of s vanishes, as also the values of the algebraic construct associated with |(5,2)=0, whose neighbour. of : for which beside f(5,5) =0 we have also of(5,3)/s=0, and also hood is given by z=c+!,s=d+Q!), there be a coefficient of rl in in general the point == 0, the roots of this equation about any point R(sx)d3d, this will be r times the coefficient of in R(s,a) or 2=c are given by H-power series in 2-c. About a finite point 2=cR[d40(1), c+M, namely will be the coefficient of rt in the sum of for which the equation af(5,2)/os = o is satisfied by one or more of the the , series obtainable from Rld 40(1). c+") by replacing ? by web roots s of f(s.3)=0, the n roots break up into a certain number of where w is an rth root of unity; thus the sum of the coefficients ? cycles, the r roots of a cycle being given by a set of power series in riin R(5,2)dz]dt for all the places which arise for z=c, and the correo a radical (s-c)', these series of the cycle being obtainable from sponding values of s, is equal to the coefficient of (3-0)- in R(51,2) + one another by replacing (2-c)" by wsz-6)*/", where w, equal to R(52,3)ť +R(sn,), where si, ... Sn are the n values of s for a exp (2x ih/r), is one of the rth roots of unity. Putting then 2-c=r value of : ncar to z=c; this latter sum ER(si, 2) is, however, a we may say that the r roots of a cycle are given by a single power rational function of z only. Similarly, ncar :=, for a place given series in 1, an increase of 2in the phase of giving an increase of by Tiar, s=d+0(1), or 5-1= Q), the coefficient of rlin R(5,2)dade 2#r in the phase of :-c. This single series in l, giving the values of is equal to -4 times the coefficient of in Rid+Q(1), 6), that is s belonging to one cycle in the neighbourhood 2=0 when the phase equal to the negative coefficient of rl in the sum of the r series of 2-c varies through 277, is to be looked upon as defining a single Rd+Q(ul), 1. so that, as before, the sum of the coefficients of place among the aggregate of values of sand s which satisfy f(s.3) = 0; in R(5,2)dade at the various places which arise for z= is equal two such places may be at the same point (8=6 sed) without to the negative coefficient of in the same rational function of coinciding, the corresponding power series for the neighbouring ER(S1,2). Thus, from the corresponding theorem for rational functions points being different. Thus for an ordinary value of 5, 2=C, there of one variable, the general theorem now being proved is seen to are n places for which the neighbouring values of s are given by » follow. power series in 2-c; for a value of $ for which af(5,2) ds =0 there are less than n places. Similar remarks hold for the neighbourhood Apply this theorem now to the rational function of sand 3, ol 3 = ; there may be n places whose neighbourhood is given by » power series in grb or fewer, one of these being associated with a R7S,3) da series in t, where 1 = ()%; the sum of the values of r which thus at a zero of R(s, 2) near which R(s, 3) =H(), we have arise is always n. In general, then, we may say, with it of one of the forms (s-c). (2-), , ()', that the neighbourhood of dR(s.s) dz any place (c,d) for which (c,d) o is given by a pair of expressions where a denotes the generalized logarithmic function, that is equal R(5,2) diai-R(s,:))). s=c+P(1), s=d+Q(!), where P(!) is a (particular case of a) power series vanishing for 1+0, and Q(1) is a power series vanishing for mtlt power series in t; leo, and I vanishes at (c,d), the expression 2-c being replaced by similarly at a place for which R(s, z)=r*K(); the theorem l when c is infinite, and similarly the expression s-d by od when d is infinite. The last case arises when we consider the finite values of 2 for which the polynomial coefficient of guna vanishes. Of such a R(5,) pair of expressions we may obtain a continuation by writing i=+ thus gives Em=&y, or, in words, the total number of zeros of R(s,s) Air +13p+ ., where is a new variable and is not zero; on the algebraic construct is equal to the total nunber of its poles. in particular for an ordinary finite place this equation simply becomes the same is therefore true of the function Rés, 2)-A, where A is an 1 = 1+1. It can be shown that all the pairs of power series 2=c+ arbitrary constant; thus the number in question, being equal to the P(!), s=d+Q50) 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 f(5,2)=0 can be obtained from one that R(5,5) = A on the algebraic construct. 1 dR(s.2); to |