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as before there is a least value for», actually occurring in one or if A denote the generalized logarithm, -wλ[f(zo+w')}-λ[ƒ(%)]}, that more periods, say in the period How + row; now take, if + is, since f(s+w') = f(zo), gives 2riNw, where N is an integer; similarly be a period, N'+', where N' is an integer, and o<; the result of the integration along the other two opposite sides is of thence wμw+N' (π-μow)+'w'; take then-N'-Nλo+λ', the form 2riN'w', where N' is an integer. The integral, however, where N is an integer and is as above, and o<; we thus have a period No+N's+λ'w+v'w', and hence a period interior to the parallelogram. For a zero, of order m, of f(z) at 2=a, is equal to 2x1 times the sum of the residues of zf'(z)/f(z) at the poles A'w+v'w', wherein '<,<ro; hence o and X'=0. All the contribution to this sum is 2x1ma, for a pole of order n at z=b periods of the form w+w are thus expressible in the form the contribution is -2winb; we thus infer that Ema-Enb = Nw+N'w': No+N', where, are periods and N. N' are integers. But this we express in words by saying that the sum of the values of in fact any complex quantity, P+Q, and in particular any other where f(z) =o within any parallelogram is equal to the sum of the possible period of the function, is expressible, with u, real, in the values of z where f(z) = save for integral multiples of the periods. form + for if w=ptio, w=ptio, this requires only By considering similarly the function f(3)-A where A is an arbitrary P=μp+ve' Q=po+vo', 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 μ and v. the values of 2 where the function takes the value A in the parallelogram.

It thus appears that if a single valued monogenic function of be periodic, either all its periods are real multiples of one of them, and then all are of the form M2, where is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form No+N', where 2, are periods, and N, N' are integers. In the former case, putting212/2, and the function f(2) = (5), the function (3) has, like exp (5), the period 2ri, and if we take t=exp (5) or =λ() the function is a single valued function of . If then in particular f(s) is an integral function, regarded as a function of t, it has singularities only for t=0 and !=∞, and may be expanded in the form a Taking the case when the single valued monogenic function has wo periods w, w' whose ratio is not real, we can form a network of parallelograms covering the plane of s whose angular points are the points c+mw+m'w', wherein c is some constant and m, m' are all possible positive and negative integers; choosing arbitrarily one of these parallelograms, and calling it the primary parallelogram, all the values of which the function is at all capable occur for points of this primary parallelogram, any point, &, of the plane being, as it is called, congruent to a definite point, z, of the primary parallelogram, 2-z being of the form mw+m'w', where m, m' are integers. Such a function cannot be an integral function, since then, if, in the primary parallelogram |f(z)|<M, it would also be the case, on a circle of centre the origin and radius R, that f(s)| <M, and therefore, if Ea,2" be the expansion of the function, which is valid for an integral function for all finite values of z, we should have an<MR, which can be made arbitrarily small by taking R large enough. The function must then have singularities for finite values of z.

We pass now to the construction of a function having two arbitrary periods w, w' of unreal ratio, which has a single pole of the second order in any one of its parallelograms.

For this consider first the network of parallelograms whose corners negative integer values; putting a small circle about each corner are the points =mw+m'w', where m, m' take all positive and of this network, let P be a point outside all these circles; this will be interior to a parallelogram whose corners in order may be denoted by zo, zo+w, zo+w+w', zo+w'; we shall denote zo, zo+ by A., Bo; this parallelogram II, is surrounded by eight other parallelograms, forming with II, a larger parallelogram I, of which one side, for instance, contains the points zo-w-w', zo-w', zo-w'+w, zo-w'+2w. which we shall denote by A, B, C, D. This parallelogram II, is surrounded by sixteen of the original parallelograms, forming with a still larger parallelogram I, of which one side, for instance, contains the points 20-2w-zw', 20-w-zw', zo-2w, zo+w-zw' 20+2w-zw, zo+3w-2w', which we shall denote by A2, B2, C2, D2. the distances of the point P from the corners of all the original E, F. And so on. Now consider the sum of the inverse cubes of parallelograms. The sum will contain the terms

I

So=PA3+ (PA1+PB; +PC) + (PA++...+PĖ;).
+PE) +...

and three other sets of terms, each infinite in number, formed in a
similar way. If the perpendiculars from P to the sides AB.
A,B,C, A,B,C,D,E,, and so on, be p. p+q, p+2g and so on, the
sum So is at most equal to

the series

3

5
(p+29)

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We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram, 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. Supposing the constant c used in naming the corners of the parallelo-equality with 29-n, so that the series So is convergent, as we know grams so chosen that no pole falls on the perimeter of a parallelogram, the sum En to be, this assumes that po; if P be on A,B the proof for the convergence of S-1/PA is the same. Taking it is clear that the integral f(z)dz round the perimeter of the the three other sums analogous to So we thus reach the result that primary parallelogram vanishes; for the elements of the integral corresponding to two such opposite perimeter (points as 2, z+w (or as s, z+w') are mutually destructive. This integral is, however, equal to the sum of the residues of f(z) at the poles interior to the parallelogram. Which sum is therefore zero. There cannot therefore be such a function having only one pole of the first order in any parallelogram; we shall see that there can be such a function with two poles only in any parallelogram, each of the first order, with residues whose sum is zero, and that there can be such a function with one pole of the second order, having an expansion near this pole of the form (s-a)+(power series in 2-a).

Considering next the function (z) = [f(z)]-1(2), it is easily seen that an ordinary point of f(z) is an ordinary point of 4(2), that a zero of order m for f(z) in the neighbourhood of which f(z) has a form, (2-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 (2) of residue -n; manifestly (2) has the two periods of f(z). We thus infer, since the sum of the residues of (2) is zero, that for the function f(z), 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 briefly expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function f(z)-A, where A is an arbitrary constant, we have the result, that the function f(z) assumes the value A in one of the parallelograms as many times as it becomes infinite. Thus, by what is proved above, every conceivable complex value does arise as a value for the doubly periodic function f(2) in any one of its parallelograms, and in fact at least twice. The number of times it arises is called the order of the function; the result suggests a property of rational functions.

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where is mw+m'w', and m, m' are to take all positive and negative integer values, and z is any point outside small circles described with. the points as centres, is absolutely convergent. Its sum is therefore independent of the order of its terms. By the nature of the proof, which holds for all positions of 2 outside the small circles spoken of, the series is also clearly uniformly convergent outside these circles. Each term of the series being a monogenic function of 2, the series may therefore be differentiated and integrated outside these circles, and represents a monogenic function. It is clearly periodic with the periods w, w; for (z+w) is the same sum as (3) with the terms in a slightly different order. Thus (z+w)=(z) and (z+w') = P(=). Consider now the function

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I

I

wherein is a sum excluding the term for which mo and m'≈o. Hence f(z+w)-(3) and (z+w-(2) are both independent of Noticing, however, that, by its form, f(s) is an even function of a and putting 2=-w, z=-w' respectively, we infer that also fis) has the two periods w and w. In the primary parallelogram II. however, f(z) is only infinite at zo in the neighbourhood of which its expansion is of the form + (power series in 2). Thus f(2) is such a doubly periodic function as was to be constructed, having in

Consider further the integral (dz, where f(z) =), taken any parallelogram of periods only one pole, of the second order.

dz'

round the perimeter of the primary parallelogram; the contribution to this arising from two opposite perimeter points such as z and s+w is of the form-w

dz, which, as z increases from zo to zo+w, gives,

It can be shown that any single valued meromorphic function of z with w and w' as periods can be expressed rationally in terms where A, B are constants. of f(z) and (z), and that [(z)]2 is of the form 4[ƒ(2)}3+Aƒ(z)+B,

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functions f(z+1), *(2+1), which are such doubly periodic function of zas have been discussed, can each be expressed, so far as they depend on 2, rationally in terms of f(z) and (2), and therefore, so far as they depend on and , rationally in terms of f(z), f(t), ø(2) and (4).

and hence, if Z'go, since E'-(-)o, we have, for sufficiently It can in fact be shown, by reasoning analogous to that given above, small s greater than zero, — $(1)

and

f(2)=x+308.812+500.8°+..

*(3) =~28*+602.2+200, 83+...; using these series we find that the function

F(2) = [(z)]2 —4[ƒ(2)]3 +6002f(3)+1400s contains no negative powers of z, being equal to a power series in beginning with a term in s. The function F(z) is, however, doubly periodic, with periods w, w', and can only be infinite when either f(2) or (2) is infinite; this follows from its form in f(z) and (2); thus in one parallelogram of periods it can be infinite only when 2=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 it is a constant, and therefore necessarily always zero. Putting therefore f(z) =5 and (z) =d5/ds we see that

dz

=(45-60035-14001).

Historically it was in the discussion of integrals such as

fd5(45-600.5-1400),

that

J(z+1)+S(2) +ƒ(1) = 3 [† (2) = "9]

This shows that if F(2) be any single valued monogenic function which is doubly periodic and of meromorphic character, then (3+) is an algebraic function of F(z) and F(). Conversely any which is such that F(z+1) is an algebraic function of F(z) and F(1), single valued monogenic function of meromorphic character, F(z), can be shown to be a doubly periodic function, or a function obtained from such by degeneration (in virtue of special relations connecting further the fundamental differential equation is usually written The functions f(z), ø(2) above are usually denoted by P(z), B′(z); (B'z)=4(V)-g1PB-gs,

the fundamental constants).

and the roots of the cubic on the right are denoted by ez, ez, esi for the odd function, B's, we have, for the congruent arguments -w and w, B'(}w) = — B'(-}w) = B'(w), and hence B')=0; hence we can take ew), e2 = B( } w¬- Jw'), e; = B(jw'). It can then be proved that B()-e][B (s + {w)−e1] = (e1-es) (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 z=1,...zam of a parallelogram, these being, for simplicity of explanation, supposed to be all of the first order, is rationally expressible in terms of and f(z), and we proceed as follows:Consider the expression

(5.1)m+n(5,1)m-1

(x)

❖(z) = (5—A1) (5 —A1) (-A) where A.-f(a.), is an abbreviation for f(z) and for 4(2), and m and m-2, so that there are 2m unspecified, homogeneously (5.1), (5,1)-2, denote integral polynomials in 5, of respective orders entering, constants in the numerator. It is supposed that no one of the points a,...a is one of the points ma+m'w' where f(z) = The function (2) is a monogenic function of z with the periods w, w, becoming infinite (and having singularities) only when (1) or (2) one of the factors -A, is zero. In a period parallelogram including zo the first arises only for zo; since for o, is in a finite ratio to 3/2; the function (2) for = is not infinite provided the coefficient of in (5,1) is not zero; thus (2) is regular about z=0. When -A, 0, that is f(z) =f(a.), we have ===G,+mw+m'w', and no other values of 2, m and m' being integers; suppose the unspecified coefficients in the numerator so taken that the numerator vanished to the first order in each of the m points -a, -a,...am; that is, if (a,) = B., and therefore (-a) --B., so that we have the m relations

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larly the function B(z)-; like () it has a pole of the second order at =0, its expansion in its neighbourhood being of the form 'z ̃ ̄2(1-c182+Az1+...); 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−1w)B'(w)+1 (z−jw)2®® (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 [B(z)-e), if we attach a definite sign to it for some particular value of 2, is a single valued function of 2; 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) = ± fi(2), f1 (z+w') = ±ƒ1(z); it can be seen by considerations of continuity that the right sign in either of these equations does not vary with ; not both these signs can be positive, since the function has only one pole, of the first function, and hence (w)=-fi(w), which is not zero since order, in a parallelogram (w, w'); from the expansion of fi(2) about z=o, namely 1(1 − }e1s2 + ...), it follows that fi(2) is an odd [ƒ1(}w'))2=es-e, so that we have fi(z+w') = -fi(s); an equation (z+w)=-fi (2) would then give fi(z+w+w)=fi(2), and hence f(w+w') = fi(-w-w'), of which the latter is fi(w+w'); this would give fiw+1)=0, while [fi(w+w'))2= eyes. infer that fi(z+w)=fi(2), !fi (z+w') = -fi(2), f1(z+w+w') = -fi (2); The function f(z) is thus doubly periodic with the periods and 2; in a parallelogram of which two sides are w and 2w it has poles at z=0, z=w each of the first order, and zeros of the first order at z=w, z=\w+w'; it is thus a doubly periodic function of the second order with two different poles of the first order in its (2) = [B(2)—e2}+, ƒ1(2) =IB(z)-es]; they give We may similarly consider the functions parallelogram (w, 2w').

We thus

}2(3+w+w) =ƒ2(2), ƒ2(3+w) = −ƒ2(z), f2(z+w') = −ƒ2(3), fs(z+w')=ƒ23, ƒs(2+w) = −ƒ3(E), ƒs(z+w+w') = −fa(2). Taking u=2(e-es), with a definite determination of the constant (es-e), it is usual, taking the preliminary signs so that for s=0 each of sfi(2), zfa(z), zfa(z) is equal to +1, to put sn(u) = (G–e3), cn(u) = {1(2), dn(u) =}; (2) f: (=)

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k2=(exes)/(es-es), K=jw(e1-es)i, iK'=4w'(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+iK', and zeros of the first order at u=0, u=2K, similarly cn(u), dn (u)are even doubly periodic functions whose periods can be written down, and sn2(u)+cn' (u) = 1, k2sn2 (u) +dn2 (u) = 1; if x= sn(u) we at once find, from the relations given here, that du

√x = [(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 z=0 is a pole of F(2), 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),+n(3,1) (-A1)... (5-Am)

where h, k are such that the greater of 2h-2m, 2k+3-2m is equal tor; the case where some of the poles a....am are multiple is to be met by introducing corresponding multiple factors in the denominator 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

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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(%) of ▲ am(u), where 4(6) meant (1-k2 sin2 ). The addition equation for each of the functions fi(z), fi(z). f.(z) is very simple, being

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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 OR/dz =Ŕ/ot; this will require the use of the differential equation [fi]=[}(t)+er-ea][ƒ? (=) +er-ea],

and in fact we find

a

(323 − 2) log [ƒ(s) +ƒ(1) =L^(2)-f2(1) = log [f(z)-f(!)]; hence it will follow that R is a function of +t, and R is at once seen to reduce to f(s) when t=0. From this the addition equation for each of the functions sn(u), cn(u), dn(u) can be deduced at once; if S1, C1, di, S2, C2, da denote respectively sn(u), cn(u), dn(); sn(u), cn(u), dn(), they can be put into the forms

where

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The introduction of the function fi(2) is equivalent to the introduction of the function B(z; w, 2w) constructed from the periods w, 2w as was B(z) from w and w; denoting this function by B(z) and its differential coefficient by B'1(z), we have in fact fi(z) = }}}, (w3) — $1 (2) B'(3) as we see at once by considering the zeros and poles and the limit of 2f(2) when z=0. In terms of the function () the original function B(2) is expressed by

P(z) = P1(8)+P1(z+w')−B1 (w'),

as a consideration of the poles and expansion near so will show.
A function having 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-
sion me+m'w', in which m, m' are integers, is a constant multiple of
{P[z-}(a'+b')]~B{a'— }(a'+b')}} / {V[z− }(a+b)}-P[a-}(a+b)}} ;
if the expansion of this function near za be
λ(2-0)`1+μ+2μn(x−a)”,

the expansion near z=b is

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−λ(3−b)~'+u+_Σ (− 1)"μn (2—b)",

8-1

as we see by remarking that if -b=-(-a) the function has the same value at a and; hence the differential equation satisfied by the function is easily calculated in terms of the coefficients in the expansions.

From the function P(2) we can obtain another function, termed the Zeta-function; it is usually denoted by (z), and defined by

5(2)− ¦ - S: ['b - B(=) ] dz = ×· (——++)·

for which as before we have equations

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}(2+w)=5(2) + 2xin, 5(z+w')=3(2)+2xin', where 2n, 27' are certain constants, which in this case do not both vanish, since else (2) would be a doubly periodic function with only one pole of the first order. By considering the integral

(2)dz

round the perimeter of a parallelogram of sides w, w containing z=o in its interior, we find nw'n'w = 1, so that neither of n, n' is zero. We have '(x)=-P(3). From (2) by means of the equation

(2)

2

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=exp

periodic function without poles, that is, a constant; this gives the expression of F(s) referred to. The indefinite integral fF(2)dz can then be expressed in terms of z, functions P(z-a) and their differential coefficients, functions (2-a) and functions log (z-a).

§ 15. Potential Functions. Conformal Representation in of existence of a single valued monogenic function, u+iv, of General. Consider a circle of radius a lying within the region the complex variable z, =x+iy, the origin so being the centre of this circle. If z=rE(ip)=r(cos +i sin ø) be an internal point of this circle we have

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and the equivalence of the integrals holds for every arc of of which the second part does not depend upon the position of z, integration.

and describe a small circle with centre at P, cutting the given circle in Conversely, let U be any continuous real function on the circumference, U. being the value of it at a point P, of the circumference, A and B, so that for all points P of the arc APB we have U-Ul<, 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 Q be restricted to lie in the small space bounded by the arc A'P.B' and this second circle, then for all positions of P upon the greater arc AB of the original circle QP is greater than a definite integral finite quantity which is not zero, say QP> D Consider now the (a2-r) +r2-2arcos (0·

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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 = 9; it can be
seen to satisfy the equations

σ(z+w)

(2)

=-exp [2rin(s+}w)],

o(s+w') σ(2)

·-exp{2xin'(3+}w')]. +am=a'iPa's+..

By means of these equations, if atat

ta'm, it is readily shown that

...

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is a doubly periodic function having a....am as its simple poles, and a',... a'm as its simple zeros. Thus the function (3) has the important property of enabling us to write any meromorphic doubly periodic function as a product of factors each having one zero in the parallelogram of periods; these form a generalization of the simple factors, -a, which have the same utility for rational functions of z We have (2)=0′(s)/o (3);

The functions (2), (3), may be used to write any meromorphic doubly periodic function F() as a sum of terms having each only one pole; for if in the expansion of F(z) near a pole s=a the terms with negative powers of s-a be

A1(-a)¬1+A,(-a)2+...+Am+1(8-a)−(m+1),

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-USAP.B (U-Uo)dwSAP, (U-U)d0+

2

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"If the finite angle between QA and QB be called and the finite angle AOB be called e, the sum of the first two components is numerically less than

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the last component is numerically less than
If the greatest value of 1 (U-U) on the greater arc AB be called H,

H

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of which, when the circle, of centre Po, passing through A'B' is
appears that u' is a function of the position of Q whose limit, when Q.
sufficiently small, the factor a is arbitrarily small. Thus it
U. From the form
interior to the original circle, approaches indefinitely near to Po, is

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since the inclination of QP to a fixed direction is, when Q varies, P remaining fixed, a solution of the differential equation

+

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of the same form, one for each of the poles in a parallelogram of | where s,=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 (a-ri)

where

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de

-SU [1+2 cos(0-4)+2 cos 2 (0-4)+... ..]do

=a+ax+by+q(x2-y2)+2b2xy+...

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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 2/dx2+ð2v/áy2; the series is thus the real part of a power series in z, 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 xo. yo of this region by a power series in x-xo, y-yo, 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 a2p, ap

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= 0,

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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. §. y.... ben real numbers, whose sum is n-2, the integral

as describes the real axis, describes in the plane of a polygon of n s=f(l-a)a-1(l—b)ß-1...di, 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, t=1, t=0.

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

[(+h2)/(1‍+h222)]*, with 1 when s=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 sih; if we plot the value of upon another plane, as z describes the continuous curve ABCDE will describe the real axis from 1 to 5=-1, the point C giving the expansion of } is 5-k=222+...

Suppose in particular, e being any point interior to Re, 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=o, and the points B, D giving the points h. Near z=0 perimeter of R. Then the function of z expressed by or s+h-22+...

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

will be developable by a power series in (2-2) about every point zo interior to Ro, and will vanish at z=c; while on the boundary of R 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 with centre at =0, this latter point corresponding to z=c. A closed path within Ro. passing once round 2=c, will lead to a closed path passing once about $0. Thus every point of the interior of R will give rise to one point of the interior of the circle. The converse is also truc, but is more difficult to prove; in fact, the differential coefficient dy/dz does not vanish for any point interior to R. This being assumed, we obtain a conformal representation of the interior of the region R upon the interior of a circle, in which the arbitrary interior point c of R corresponds to the centre of the circle, and, by utilizing the arbitrary constant arising in determining the function Q, an arbitrary point of the boundary of R corresponds to an arbitrary point of the circumference of the circle.

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 Udo-Ude 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

1-44

in either case an increase of in the phase of z gives an increase of in the phase of -h or 5+h. Near z=ih the expansion of is =(-ih) (2ih/(1-4)+..., and an increase of 2 in the phase of z-ih also leads to an increase of in the phase of . Then as z describes the semicircle EFA, also describes a semicircle of radius unity, the point = becoming =i. There is thus a conformal representation of the interior of the slit semicircle in the z-plane, upon the interior of the whole semicircle in the 5-plane, the function z=[(52-h2)/(1-h252)]}

being single valued in the latter semicircle. By means of a transformation = (5+1)?/(5−1)2, the semicircle in the plane of can further be conformably represented upon the upper half of the whole plane of t.

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

dt 2(3-1)

dt

4w' = Sex 2(13-1)*

drawing the equilateral triangle whose vertices are O, of argument Q. 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 (2w+w'), and D, of argument (w+2w), 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=eue(-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-e2v, w+w' teu, wtw-v, w+w'-", w+w'+ev, w'-eu, w' + e2v, -e3u. Further, when u is real, since the term -2(u+mw+m'e2w)→3, which is the conjugate complex of −2 (u+mw+m'ew), arises in the infinite sum which expresses B'(), namely as -2(u+uw tμew)-3, where μ=m-m', u'=-m', it follows that B'() 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 v=euo, B'(v) is the conjugate complex of '(x). Hence it follows that the variable &=jiV'(u)

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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

u-tif,® (P-1)-Idi

in accordance with the general theory.

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

§ 15. 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 z-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; 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 case s, 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 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 2-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)}', and a series in 1/2 to a series in z-c, when z is near enough to c, by means of

(1+)

of them by this process of continuation, a fact which we express by saying that the equation f(s,z) =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.2)=0, may be considered in the neighbourhood of any place (c,d) by substituting therein =c+P(), s=d+Q(t); the result is necessarily of the form H(0). where H() is a power series in not vanishing for 0 and m is an integer. If this integer is positive, the function is said to vanish to order m at the place; if this integer is negative,, the function is infinite to order at the place. More generally, if A be an arbitrary constant, and, near (c,d), R(5,2)-A is of the form "H(1). where m is positive, we say that R(s,2) becomes m times equal to A at the place; if R(s,z) is infinite of order at the place, so also is R(s,)-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, 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 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.

That the total number of zeros of R (s,z) is finite is at once obvious, these values being obtainable by rational elimination of s between f(s,2)=0, R(s,z)=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, s, =c+P(), s=d+Q(4), expand the product which are connected by f(s,z)=0: about any place (c,d) for which

R(5,2)

in powers of t 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(S)

The theorem holds for the case n=1, that is, for rational functions

of one variable ; in that case, about any finite point we have
2-C-1, and about z=∞ we have 1, and therefore dz/dt =-12;
in that case, then, the theorem is that in any rational function of s,
A, A2
Am

Σ (A + (Ab); + ... + (A) =) +P+Qs++...+R,

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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 ; an obvious result. In general, if for a finite place of the algebraic construct associated with f(s,z) =o, whose neighbourhood is given by z=c+r,s=d+Q), there be a coefficient of in R(s,z)dz/dt, this will be r times the coefficient of in R(s,a) or R[d+Q(!), c+}, namely will be the coefficient of in the sum of the series obtainable from R[d+Q(1), c+] by replacing by wy where is an rth root of unity; thus the sum of the coefficients of in R(s,z)dz/dt for all the places which arise for z=c, and the corre sponding values of s, is equal to the coefficient of (s—c)1 in R($1,2)+ R(52,)+ +R(S), where si,... s, are the n values of 's for a value of a near to z=c; this latter sum ER(s., 2) is, however, a rational function of 2 only. Similarly, near zo, for a place given by z, s=d+Q(t), or sQ), the coefficient of in R(s,e)dz/di is equal to -r times the coefficient of in R[d+Q), ), that is equal to the negative coefficient of 21 in the sum of the r series R[d+Q(wt), ], so that, as before, the sum of the coefficients of in R(s,z)dz/dt at the various places which arise for zo is equal to the negative coefficient of in the same rational function of 2, ER(s,, z). Thus, from the corresponding theorem for rational functions of one variable, the general theorem now being proved is seen to follow.

The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation f(s,2) = Pos" + P1s"-1+. +p=0, wherein po, Pi.... p, are integral polynomials in s. Assuming f(s,z) incapable of being written as a product of polynomials rational in s and z, and excepting values of s for which the polynomial coefficient of s vanishes, as also the values of 2 for which beside f(s,z) =o we have also aƒ(s,z)/as =o, and also in general the point z=o, the roots of this equation about any point 2=c are given by power series in 2-c. About a finite point z=c for which the equation aƒ(s,2)/ds =0 is satisfied by one or more of the roots s of f(s,z) =0, the n 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 (2-c), these series of the cycle being obtainable from one another by replacing (2-c)' by w(z-c)'', where w, equal to exp (2wih/r), is one of the rth roots of unity. Putting then z-c=r we may say that the roots of a cycle are given by a single power series in t, an increase of 2 in the phase of giving an increase of 2r in the phase of z-c. This single series in 1, giving the values of s belonging to one cycle in the neighbourhood of ac when the phase of -c varies through 27, is to be looked upon as defining a single | place among the aggregate of values of z and s which satisfy f(s,2) = 0; two such places may be at the same point (2=c, s=d) without coinciding, the corresponding power series for the neighbouring points being different. Thus for an ordinary value of z, s=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,2)/ds=o there are less than n places. Similar remarks hold for the neighbourhood of z=0; 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 (1); the sum of the values of r which thus arise is always n. In general, then, we may say, with t of one of the forms (c), (z—c)1/", &, (1), 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 z=c+P(1), s=d+Q), where P() is a (particular case of a) power series vanishing for to, and Q() is a power series vanishing for

Apply this theorem now to the rational function of s and 8,

I

dR(5, 2); R(5,3)

da

at a zero of R(s, z) near which R(s, z) = tTMH(!), we have
dR(s. s) dz_d
R(s,z) dz

to

T

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mt+power series in t;

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1=0, and vanishes at (c,d), the expression 2-c being replaced by similarly at a place for which R(5, 2) = K(); the theorem
when c is infinite, and similarly the expression s-d by s1 when
dis infinite. The last case arises when we consider the finite values
of for which the polynomial coefficient of s vanishes. Of such a
pair of expressions we may obtain a continuation by writing = b+
ALT+AST2+..., where is a new variable and A, is not zero;
in particular for an ordinary finite place this equation simply becomes
14+. It can be shown that all the pairs of power series =c+
P), s=d+Q) which are necessary to represent all pairs of values
of z, s satisfying the equation f(s,z) o can be obtained from one

thus gives Em=Eu, or, in words, the total number of zeros of R(s, a)
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,) A on the algebraic construct.

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