of centre the origin and radius R, that f(2)] <M, and therefore, if So=PA3+ (PA+B+pċ;) + (þæ+pģ¿+.... · +PE:) + .......

functions f(s+1), (z+1), which are such doubly periodic function of sas have been discussed, can each be expressed, so far as they depend on 2, rationally in terms of f(z) and (z), and therefore, so far as they depend on s and t, rationally in terms of f(z), ƒ(t), ø(z) and (1).

and hence, if 'to, since '(-1)=0, we have, for sufficiently It can in fact be shown, by reasoning analogous to that given above, small greater than zero,

and

f(2)=2+301.22+508.8°+..

(s)=-22+60;.s+200; s3+...; using these series we find that the function

F(2) =[6(2)]2 —4[ƒ(2)]'+6002f(z)+14003 contains no negative powers of 2, being equal to a power series in 2 beginning with a term in s. The function F(2) is, however, doubly periodic, with periods w, w', and can only be infinite when either f(z) or (2) is infinite; this follows from its form in f(s) and (2); thus in one parallelogram of periods it can be infinite only when z=0; we have proved, however, that it is not infinite, but, on the contrary, vanishes, when zo. Being, therefore, never infinite for finite values of z it is a constant, and therefore necessarily always zero. Putting therefore ƒ(a) =} and ø(z) =d5/dz we see that dz

(45-60025-14001)-4.

Historically it was in the discussion of integrals such as

Jd5(45-6002.5-140)+

that

ƒ(2+1) +ƒ(3) +S(0) = 1 [$ (2) =¿ (1

This shows that if F(2) be any single valued monogenic function which is doubly periodic and of meromorphic character, then (+) is an algebraic function of F(z) and F(). Conversely any which is such that F(+) is an algebraic function of F(z) and F(1), single valued monogenic function of meromorphic character, F(2), from such by degeneration (in virtue of special relations connecting can be shown to be a doubly periodic function, or a function obtained the fundamental constants).

further the fundamental differential equation is usually written
The functions f(z), ø(2) above are usually denoted by T(z). P′(s);
(B'z)2=4(Bz)1-g: PS-83,

and the roots of the cubic on the right are denoted by e, e, esi
for the odd function, B's, we have, for the congruent arguments
-w and w, B'(}w) = − V′(−}w) = − 'B'(w), and hence B')=0;
hence we can take e1 = P(w), e2 = B ( }w -- Jw'), es = B(w). It can
then be proved that [P)-[B (≈ + }w)−e,] = (es-e1) (e-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

where sin z, it has proved finally to be simpler to regard as a function of z. We shall come to the other point of view below, under § 20, Elliptic Integrals.

To prove that any doubly periodic function F(z) with periods w, w', having poles at the points sai,...za of a parallelogram, these being, for simplicity of explanation, supposed to be all of the first order, is rationally expressible in terms of (z) and f(z), and we proceed as follows:Consider the expression

(3.1)m+n(5,1)m-3

larly the function (2)-e; like (2) it has a pole of the second order at 20, its expansion in its neighbourhood being of the form z2(1-6,22+As+..), having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (w, w'). In fact near its zero its expansion is (x−}w)P′(}w)+ } ( z − }w)1Ÿ® (Jw) + ; we have seen that B'(w)=0; thus it has a zero of the second order wherever it vanishes. Thus it appears that the square root [(2)-e], if we attach a definite sign to it for some particular value of 2, is a single valued function of ; for it can at most have two values, and the only small circuits in the plane which could lead to an interchange of these values are those about either a pole or a zero, neither of which, as we have seen, has this effect; the function is therefore single valued for any circuit. Denoting the function, for a moment, by fi(2), we have fi(z+w) = ±ƒ¡ (8), f1(2+w') = ±ƒ1(2); it can be seen by considerations of continuity that the right sign (=): = — (5—A1) (5 — A2) ....(5-Am) in either of these equations does not vary with s; not both these where A, f(a.), is an abbreviation for f(z) and for (3), and signs can be positive, since the function has only one pole, of the first (5,1), (3,1)-2, denote integral polynomials in 5, of respective orders order, in a parallelogram (w, w'), from the expansion of fi(z) about m and m-2, so that there are 2m unspecified, homogeneously function, and hence fi(-)=-fi(w), which is not zero since =o, namely 1 (1 — {e, z2 + . . .), it follows that f(z) is an odd entering, constants in the numerator. It is supposed that no one of the points a,...a is one of the points mo+m'w' where f(z) = (z+w)=-fi (2) would then give fi(z+w+w)=fi(2), and hence [f1(}w'))2=e-e, so that we have fi(+w') = f(2); an equation The function (2) is a monogenic function of z with the periods w. w. fi(w+w') = fi (- tw-}w'), of which the latter is f(w+w), this becoming infinite (and having singularities) only when (1) = ∞ or (2) one of the factors -A, is zero. In a period parallelogram infer that fi(z+w)=fi(2), Ifi(3+w) = -fi(2), fi (=+w+w') =-fi (8); would give fiw+w)=0, while [fi(w+w')2=ee. We thus including so the first arises only for zo; since for o, is in The function f(s) is thus doubly periodic with the periods w and a finite ratio to 3/2; the function (2) for = is not infinite 2: in a parallelogram of which two sides are w and 2w it has provided the coefficient of in (5.1) is not zero; thus (2) is poles at =0, s=w each of the first order, and zeros of the first regular about 2-0. When A.-o, that is f(z) =f(a.), we have z= ±a,+mw+m'w', and no other values of 2, m and m' being of the second order with two different poles of the first order in its order at z, z=w+w'; it is thus a doubly periodic function integers; suppose the unspecified coefficients in the numerator so parallelogram (w, 2w'). taken that the numerator vanished to the first order in each of the()=['B(=)-ea]), fa(z) = [V()-es; they give We may similarly consider the functions m points a, as,... - am; that is, if (a.) = B., and therefore (-a)=-B,, so that we have the m relations

F(2) = A4(2),

}:(=+w+w)=√2(2), ƒ:(s+w) = −ƒ2(2), ƒ2(x+w') = −ƒ:(z), fs(2+w')=ƒs2, fi(z+w) = −ƒ3(2), ƒs(z+w+w') = −ƒ1(s). Taking u=(-es), with a definite determination of the constant (-es), it is usual, taking the preliminary signs so that for s=0 each of sfi(2), zf:(≈), zf,(z) is equal to +1, to put (e1-es) f(2) (u)cn(u)=(2) f1(2) dn (u) =3(3)* fa(2)

sn(u) =

k2=(exes)/(ex-es), K=jw(er-es)}, iK'=}w'(er-es)}; thus sn(u) is an odd doubly periodic function of the second order with the periods 4K, 2iK, having poles of the first order at u=iK', u=2K+K', and zeros of the first order at uo, u=2K, similarly cn(u), dn (u)are even doubly periodic functions whose periods can be written down, and sn2(u)+cn2(u) = 1, k2sn2(u) +dn2(u) = 1; if x = sn(u) we at once find, from the relations given here, that

du

Tx={(1−x2)(1−k2x2)] ̄};

where A is a constant; by which F(z) is expressed rationally in if we put x=sin & we have terms of f(z) and (2), as was desired.

When zo is a pole of F (z), say of order 7, the other poles, each of the first order, being a....am, similar reasoning can be applied to a function

(5.1),+7(5.1)k
(-A)... (-A)

where h, kare such that the greater of 2h-2m, 2k+3-2m is equal
to 7; the case where some of the poles a....a are multiple is
to be met by introducing corresponding multiple factors in the de-
nominator and taking a corresponding numerator. We give a
solution of the general problem below, of a different form,
One important application of the result is the theorem that the

du

and if we call the amplitude of u, we may write =am(u), x=sin.
am(u), which explains the origin of the notation sn(u). Similarly
cn(u) is an abbreviation of cos. am(u), and dn(u) of A am(u), where
4(4) meant (1-k2 sin2 )). The addition equation for each of the
functions fi(2), fa(2), fa(s) is very simple, being
ƒ(2) +ƒ(1) _ ƒ (2)ƒ' (1) —ƒ(0)ƒ'(3),

|

we determine an integral function (), termed the Sigma-function, Hence we can write

having a zero of the first order at each of the points = 2; it can be

will not be infinite at z=a. Adding to this a sum of further terms of the same form, one for each of the poles in a parallelogram of | where 2,=x+iy, is the point Q, we infer that u' is a differentiable

function satisfying this equation; indeed, when r<a, we can write | proposed for ABC; we can then determine a function for the interior

In this series the terms of order n are sums, with real coefficients, of the various integral polynomials of dimension n which satisfy the equation /x+/ay; the series is thus the real part of a power series in 2, and is capable of differentiation and integration within its region of convergence.

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 of expression about any interior point xe. yo of this region by a power series in x-xo, y-ys, with real coefficients, these various series being obtainable from one of them by continuation. For any region Ro interior to the region specified, the radii of convergence of these power series will then have a lower limit greater than zero, and hence a finite number of these power series suffice to specify the function for all points interior to Ro. Each of these series, and therefore the function, will be differentiable; suppose that at all points of R, the function satisfies the equation

Suppose in particular, c being any point interior to Ro, that P approaches continuously, as z approaches to the boundary of R, to the value log r, where r is the distance of c to the points of the perimeter of R. Then the function of z expressed by

=(2-c) exp (-P-iQ)

of CFAB with the boundary values so prescribed. This in its turn. 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 functions, so alternately determined, have a limit representing such a potential function as is desired for the interior of the original 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. shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem.

A particular case of the problem is that of the conformal representation of the interior of a closed polygon upon the upper half of the plane of a complex variable t. It can be shown without much difficulty that if a, b, c,... be real values of t, and a, B, Y.... ben real numbers, whose sum is n-2, the integral

as describes the real axis, describes in the plane of z a polygon of n sides with internal angles equal to ar, Br,..., and, a proper sign being given to the integral, points of the upper half of the plane of t give rise to interior points of the polygon. Herein the points a, b... of the real axis give rise to the corners of the polygon; the condition Zan-2 ensures merely that the point to does not correspond. to a corner; if this condition be not regarded, an additional corner and side is introduced in the polygon. Conversely it can be shown that the conformal representation of a polygon upon the half plane 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 to be at arbitrary positions, such as t=0, 1=1, t=0.

As an illustration consider in the plane of 2,-x+iy, the portion of the imaginary axis from the origin to zik, where h is positive and less than unity; let C be this point sih; let BA be of length unity along the positive real axis, B being the origin and A the point 2=1; let DE be of length unity along the negative real axis, D being also the origin and E the point -1; let EFA be a If we put semicircle of radius unity, F being the point =i.

[(+h2)/(1+h2z2)]2, with 1 when z=1, the function is single valued within the semicircle, in the plane of z, which is slit along the imaginary axis from the origin to z=ih; if we plot the value of upon another plane, as z describes the continuous curve ABCDE, will describe the real axis from 1 to -1, the point C giving =0, and the points B, D giving the points h. Near 2=0 the expansion of 5 is 5-k=322 +.... 2h

T

or 5+h=-1-h1 2h will be developable by a power series in (3-2) about every point 20 in either case an increase of r in the phase of a gives an increase interior to Ro, and will vanish at z=c; while on the boundary of R of in the phase of -h or 5th. Near z-ih the expansion of is it will be of constant modulus unity. Thus if it be plotted upon a plane of the boundary of R will become a circle of radius unity-ih also leads to an increase of in the phase of 5. Then as 3 5=(3—ih)}{2ih/(1¬h)]+..., and an increase of 2 in the phase of with centre at =0, this latter point corresponding to z=c. A describes the semicircle EFA, also describes a semicircle of radius closed path within Re, passing once round 2=c, will lead to a closed path passing once about $-0. Thus every point of the interior of unity, the point z=i becoming =i. There is thus a conformal R will give rise to one point of the interior of the circle. The conrepresentation of the interior of the slit semicircle in the z-plane, verse is also true, but is more difficult to prove; in fact, the differ- upon the interior of the whole semicircle in the 5-plane, the function ential coefficient dy/dz does not vanish for any point interior to R. 8=[(52-h2)/(1-h252)]1 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 (5+1)/(5-1), the semicircle in the plane of can arbitrary interior point c of 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 an arbitrary point of the circumference of the circle.

I 2T

There thus arises the problem of the determination of a real monogenic potential function, single valued and finite within a given arbitrary region, with an assigned continuous value at all points of the boundary of the region. When the region is circular this ́problem is solved by the integral Ude Udw-- Ud previously given. When the region is bounded by the outermost portions of the circumferences of two overlapping circles, it can hence be proved that the problem also has a solution; more generally, consider a finite simply connected region, whose boundary we suppose to consist of a single closed path in the sense previously explained. ABCD: joining A to C by two non-intersecting paths AEC, AFC lying within the region, so that the original region may be supposed to be generated by the overlapping regions AECD, CFAB, of which the common part is AECF; suppose now the problem of determining a single valued finite monogenic potential function for the region AECD with a given continuous boundary value can be solved, and also the same problem for the region CFAB; then it can be shown that the same problem can be solved for the original area. Taking Indeed the values assigned for the original perimeter ABCD, assume arbitrarily values for the path AEC, continuous with one another and with the values at A and C; then determine the potential function for the interior of AECD; this will prescribe values for the path CFA which will be continuous at A and C with the values originally

As another illustration we may take the conformal representation of an equilateral triangle upon a half plane. Taking the elliptic function (a) for which B(u)=4(u)-4, so that, with e-exp (i), we have e1, e2=e2, e1e, the half periods may be taken to be dt jw' = 2(-1)

-Jew; drawing the equilateral triangle whose vertices are O, of argument O, A of argument w, and B of argument w+w-ew, and the equilateral triangle whose angular points are O, B and C, of argument w', let E, of argument (3w+w), and D, of argument (w+2), be the centroids of these triangles respectively, and let BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut OC, BC, OB in F, G, H respectively; then if u+in be any point of the interior of the triangle OEH and v=eu。=e(§-in) be any point of the interior of the triangle OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB, EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by -e, w+eu, w-ev, w+w' teu, wtw-v, w+w-21, wtw' tev, w'-en, w'ev, -eu. Further, when u is real, since the term 2 (u+mw+m'e3w), which is the conjugate complex of -−2 (u+mw+m'ew)-3, arises in the infinite sum which expresses B'(), namely as -2(u+w+new), where μ=m-m', μ'=-m', it follows that P'(u) is real; in a similar way we prove that '(u) is pure imaginary when u is pure imaginary, and that P'(u)=B'(eu) = P(eu), as also that for veto, P'(v) is the conjugate complex of '(u). Hence it follows that the variable =}iB'(u)

takes each real value once as u passes along the perimeter of the triangle ODE, being as can be shown respectively ∞, 1, 0, -1 at O, D, H, E, and takes every complex value of imaginary part positive once in the interior of this triangle. This leads to

น.

in accordance with the general theory.

It can be deduced that r = represents the triangle ODH on the upper half plane of r, and -(1-1) represents similarly the triangle OBD.

of them by this process of continuation, a fact which we express by. saying that the equation f(s,2) =o defines a monogenic algebraic construct. With less accuracy we may say that an irreducible algebraic equation f(s,z) =o determines a single monogenic function s of z.

Any rational function of z and s, where f(s,s)=o, may be considered in the neighbourhood of any place (c,d) by substituting therein = c + P(1), s=d+Q(); the result is necessarily of the form "H(1), integer. If this integer is positive, the function is said to vanish where H() is a power series in not vanishing for t=0 and m is an to order m at the place; if this integer is negative, μ, the function is infinite to order at the place. More generally, if A be an arbitrary constant, and, near (c,d). R(s,z)-A is of the form H(), where m is positive, we say that R(s,z) becomes m times equal to A at the place; if R(s,) is infinite of order at the place, so also is R(s,z)-A. It can be shown that the sum of the values of m at all the places, including the places z=, where R(s,2) vanishes, which we call the number of zeros of R(s,z) on the algebraic construct, is finite, and equal to the sum of the values of where R (s,z) is infinite, R(s,2)=A; this we express by saying that a rational function and more generally equal to the sum of the values of m where R(s,z) takes any value (including ) the same number of times or the algebraic construct; this number is called the order of the rational function.

§ 16. Multiple valued Functions. Algebraic Functions.-The explanations and definitions of a monogenic function hitherto given have been framed for the most part with a view to single valued functions. But starting from a power series, say in z-c, which represents a single value at all points of its circle of convergence, suppose that, by means of a derived series in 8-c, where c' is interior to the circle of convergence, we can continue the function beyond this, and then by means of a series derived from the first derived series we can make a further continuation, and so on; it may well be that when, after a closed circuit, we again consider points in the first circle of convergence, the value represented may not agree with the original value. One example is the cases, for which two values exist for any value of z; another is the generalized logarithm A(z), for which there is an infinite number of values. In such cases, as before, the region of existence of the function consists=c+P(), 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 points of the point-aggregate constituting the region of existence, are those points in whose neighbourhood the radii of convergence of derived series have zero for limit. In this description the point = does not occupy an exceptional position, a power series in z-c being transformed to a series in 1/2 when z is near enough to c by means of z-c=c(1-cz ̄ ̄1) [1-(1-cz ̄1))1, and a series in 1/2 to a series in 2-c, when z is near enough to c, by means of (1+

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 (s,2)=0, R(5,2)=0. That the number is equal to the total number of infinities is best deduced by means of a theorem which is also of more general utility. Let R(s,z) be any rational function of s, 2, which are connected by f(s,2)=0; about any place (c,d) for which

=

in powers of and pick out the coefficient of . There is only a finite number of places of this kind. The theorem is that the sum of these coefficients of is zero. This we express by

R(5,2)

The theorem holds for the case = 1, that is, for rational functions
2-ct, and about = we have 1, and therefore dz/dt=-
of one variable z; in that case, about any finite point we have
in that case, then, the theorem is that in any rational function of
A,
Am
(-a)

Σ (Act Abst...+A) +P+Q-1+...+R,

...

-a

the sum EA, of the sum of the residues at the finite poles is equal
to the coefficient of 1/2 in the expansion, in ascending powers of 1/2,
about 2; an obvious result. In general, if for a finite place
of the algebraic construct associated with f(s,z) =0, whose neighbour-
hood is given by z=c+r,s=d+Q), there be a coefficient of in
R(s,z)dedt, this will be r times the coefficient of in R(s,) or
R[d+Q(!), c+r), namely will be the coefficient of in the sum of
ther series obtainable from R[d+Q(t), c+r] by replacing by wl,
where w is an rth root of unity; thus the sum of the coefficients of
in R(s, 2)dz/d! for all the places which arise for z=c, and the corre
sponding values of s, is equal to the coefficient of (z-c) in R($1,5)+
R($2,) + +R(sa,2), where $1, ... S. are the n values of s for a
value of z near to z=c; this latter sum ZR(s,, z) is, however, a
rational function of s only. Similarly, near 2, for a place given
by zt, s=d+Q(1), or s1=Q), the coefficient of in R(s,z)dz/di
is equal to -r times the coefficient of in R[d+Q(0), l, that is
equal to the negative coefficient of in the sum of the series
Rid+Q(wt), 1, so that, as before, the sum of the coefficients of
Fin Rs,z)de/di at the various places which arise for = is equal
to the negative coefficient of in the same rational function of z,
ER(s, z). Thus, from the corresponding theorem for rational functions
of one variable, the general theorem now being proved is seen to
follow.
Apply this theorem now to the rational function of s and 2,
I _dR(s,s);
R(3,2) dz

The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation f(s,z) · Pos" + pis"-1+...+p=0, wherein po. Pi.... P. are integral polynomials in s. Assuming f(s,) incapable of being written as a product of polynomials rational in s and, and excepting values of z for which the polynomial coefficient of vanishes, as also the values of z for which beside f(s,z) =o we have also of(s,z)/as=o, and also in general the point zo, the roots of this equation about any point =c are given by power series in 2-c. About a finite point =c | for which the equation aƒ(5,2)/əs=o is satisfied by one or more of the roots s of f(s,2)=0, then roots break up into a certain number of cycles, the roots of a cycle being given by a set of power series in a radical (3-c)', these series of the cycle being obtainable from one another by replacing (s-c)' by w(2−c)'', where w, equal to exp (2ih/r), is one of the rth roots of unity. Putting then 2-c= we may say that the r roots of a cycle are given by a single power series in t, an increase of 27 in the phase of giving an increase of 27 in the phase of 2-c. This single series in 1, giving the values of s belonging to one cycle in the neighbourhood of a=c when the phase of -c varies through 277, is to be looked upon as defining a single place among the aggregate of values of z and s which satisfy f(s,z) = 0; | two such places may be at the same point (s=c, s=d) without coinciding, the corresponding power series for the neighbouring points being different. Thus for an ordinary value of z, z=c, there are n places for which the neighbouring values of s are given by n power series in 2-c; for a value of a for which af(s,z)/ds=o there are less than n places. Similar remarks hold for the neighbourhood of zo; there may be n places whose neighbourhood is given by n power series in or fewer, one of these being associated with a series in , where =(); the sum of the values of r which thus arise is always n. In general, then, we may say, with of one of the forms (2-c), (2−c)1/", z', (), that the neighbourhood of 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 8=c+P(!), s=d+Q(t), where P(t) is a (particular case of a) power series vanishing for 1-0, and Q() is a power series vanishing for =0, and vanishes at (c,d), the expression -c being replaced by similarly at a place for which R(s, 2) = when c is infinite, and similarly the expression s-d by s1 when

d is infinite. The last case arises when we consider the finite values of for which the polynomial coefficient of vanishes. Of such a pair of expressions we may obtain a continuation by writing

.., where is a new variable and A is not zero; in particular for an ordinary finite place this equation simply becomes +r. It can be shown that all the pairs of power series z=c+ P(1), s=d+Q(t) which are necessary to represent all pairs of values of a, s satisfying the equation f(s,z) =o can be obtained from one

at a zero of R(s,z) near which R(s, z) = H(1), we have
- I
dR(s, 2) dzd

to

R(s,z)

mr+power series in t;

thus gives Em=Eu, or, in words, the total number of zeros of R(s, 2) on the algebraic construct is equal to the total number of its poles. The same is therefore true of the function R(s, 2)-A, where A is an arbitrary constant; thus the number in question, being equal to the number of poles of R(s,z)-A, is equal also to the number of times that R(s, e) A on the algebraic construct.