Part 30

as t describes the real axis, describes in the plane of z a polygon of n sides with internal angles equal to [alpha][pi], [beta][pi], ..., 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 [Sigma][alpha] = n - 2 ensures merely that the point t = [oo] 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 = [oo].

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, D being also the origin and E the point z = -1; let EFA be a semicircle of radius unity, F being the point z = i. If we put [zeta] = [(z^2 + h^2)/(1 + h^2z^2)]^1/2, with [zeta] = 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 [zeta] upon another plane, as z describes the continuous curve ABCDE, [zeta] will describe the real axis from [zeta] = 1 to [zeta] = -1, the point C giving [zeta] = 0, and the points B, D giving the points [zeta] = [+-]h. Near z = 0 the expansion of [zeta] is [zeta] - h = z^2 1 - h^4 / 2h + ..., or [zeta] + h = -z^2 (1 - h^4)/2h + ...; in either case an increase of 1/2[pi] in the phase of z gives an increase of [pi] in the phase of [zeta] - h or [zeta] + h. Near z = ih the expansion of [zeta] is [zeta] = (z - ih)^1/2 [2ih/(1 - h^4)]^1/2 + ..., and an increase of 2[pi] in the phase of z - ih also leads to an increase of [pi] in the phase of [zeta]. Then as z describes the semicircle EFA, [zeta] also describes a semicircle of radius unity, the point z = i becoming [zeta] = 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 [zeta]-plane, the function

z = [([zeta]^2 - h^2) / (1 - h^2[zeta]^2)]^1/2

being single valued in the latter semicircle. By means of a transformation t = ([zeta] + 1)^2/([zeta] - 1)^2, the semicircle in the plane of [zeta] 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 RN(u) for which RN'^2(u) = 4RN^3(u) - 4, so that, with [epsilon] = exp (2/3[pi]i), we have e1 = 1, e2 = [epsilon]^2, e3 = [epsilon], the half periods may be taken to be _ _ / [oo] dt / [oo] dt 1/2[omega] = | --------------, 1/2[omega]' = | -------------- = 1/2[epsilon][omega]; _/ 1 2(t^3 - 1)^1/2 _/ e3 2(t^3 - 1)^1/2

drawing the equilateral triangle whose vertices are O, of argument O, A of argument [omega], and B of argument [omega] + [omega]' = -[epsilon]^2[omega], and the equilateral triangle whose angular points are O, B and C, of argument [omega]', let E, of argument {1/3}(2[omega] + [omega]'), and D, of argument 1/3([omega] + 2[omega]'), 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 = [xi] + i[eta] be any point of the interior of the triangle OEH and v = [epsilon]u0 = [epsilon]([xi] - i[eta]) 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 -[epsilon]v, [omega] + [epsilon]u, [omega] - [eta]^2v, [omega] + [omega]' + [epsilon]^2u, [omega] + [omega]' - v, [omega] + [omega]' - u, [omega] + [omega]' + [epsilon]v, [omega]' - [epsilon]u, [omega]' + [epsilon]^2v, -[epsilon]^2u. Further, when u is real, since the term -2(u + m[omega] + m'[epsilon]^2[omega])^(-3), which is the conjugate complex of -2(u + m[omega] + m'[epsilon]^2[omega])^3, arises in the infinite sum which expresses RN'(u), namely as -2(u + [mu][omega] + [mu]'[epsilon][omega])^(-3), where [mu] = m - m', [mu]' = -m', it follows that RN'(u) is real; in a similar way we prove that RN'(u) is pure imaginary when u is pure imaginary, and that RN'(u) = RN'([epsilon]u) = RN'([epsilon]^2u), as also that for v = [epsilon]u0, RN'(v) is the conjugate complex of RN'(u). Hence it follows that the variable

t = 1/2iRN'(u)

takes each real value once as u passes along the perimeter of the triangle ODE, being as can be shown respectively [oo], 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 _ / [oo] u = 1/3i | (t^2 - 1)^(-2/3) dt _/ t

in accordance with the general theory.

It can be deduced that [tau] = t^2 represents the triangle ODH on the upper half plane of [tau], and [zeta] = {i-[tau]^(-1)}^(1/2) represents similarly the triangle OBD.

S 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 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; 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 case z^(1/2), for which two values exist for any value of z; another is the generalized logarithm [Lambda] (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 z = [oo] does not occupy an exceptional position, a power series in z - c being transformed to a series in 1/z 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/z to a series in z - c, when z is near enough to c, by means of

1 1 / z - c \^(-1) -- = -- ( 1 + ----- ). z c \ c /

The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation [f](s, z) = p_0s^n + p1s^(n - 1) + ... + p_n = 0, wherein p0, p1, ... p_n are integral polynomials in z. Assuming [f](s, z) incapable of being written as a product of polynomials rational in s and z, and excepting values of z for which the polynomial coefficient of s^n vanishes, as also the values of z for which beside [f](s, z) = 0 we have also (Pd)f(s, z)/(Pd)s = 0, and also in general the point z = [oo], the roots of this equation about any point z = c are given by n power series in z-c. About a finite point z = c for which the equation (Pd)f(s, z)/(Pd)s = 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 r roots of a cycle being given by a set of power series in a radical (z - c)^(1/r), these series of the cycle being obtainable from one another by replacing (z - c)^(1/r) by [omega](z - r)^(1/r), where [omega], equal to exp (2[pi]ih/r), is one of the rth roots of unity. Putting then z - c = t^r we may say that the r roots of a cycle are given by a single power series in t, an increase of 2[pi] in the phase of t giving an increase of 2[pi]r in the phase of z - c. This single series in t, giving the values of s belonging to one cycle in the neighbourhood of z = c when the phase of z-c varies through 2[pi]r, 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_ (z = 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 z-c; for a value of z for which (Pd)f(s, z)/(Pd)s = 0 there are less than n places. Similar remarks hold for the neighbourhood of z = [oo]; there may be n places whose neighbourhood is given by n power series in z^(-1) or fewer, one of these being associated with a series in t, where t = (z^(-1))^(1/r); 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 (z-c), (z-c)^(1/r), z^(-1), (z^(-1))^(1/r). that the neighbourhood of any place (c, d) for which [f](c, d) = 0 is given by a pair of expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular case of a) power series vanishing for t = 0, and Q(t) is a power series vanishing for t = 0, and t vanishes at (c, d), the expression z-c being replaced by z^(-1) when c is infinite, and similarly the expression s-d by s^(-1) when d is infinite. The last case arises when we consider the finite values of z for which the polynomial coefficient of s^n vanishes. Of such a pair of expressions we may obtain a continuation by writing t = t0 + [lambda]1[tau] + [lambda]2[tau]^2 + .. , where [tau] is a new variable and [lambda]1 is not zero; in particular for an ordinary finite place this equation simply becomes t = t0 + [tau]. It can be shown that all the pairs of power series z = c + P(t), s = d + Q(t) which are necessary to represent all pairs of values of z, s satisfying the equation [f](s, z) = 0 can be obtained from one of them by this process of continuation, a fact which we express by saying that the equation [f](s, z) = 0 defines a _monogenic algebraic construct_. With less accuracy we may say that an irreducible algebraic equation [f](s, z) = 0 determines a single monogenic function s of z.

Any rational function of z and s, where [f](s, z) = 0, may be considered in the neighbourhood of any place (c, d) by substituting therein z = c + P(t), s = d + Q(t); the result is necessarily of the form t^m H(t), where H(t) is a power series in t not vanishing for t = 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, = -[mu], the function is infinite to order [mu] at the place. More generally, if A be an arbitrary constant, and, near (c, d), R(s, z) - A is of the form t^mH(t), where m is positive, we say that R(s, z) becomes m times equal to A at the place; if R(s, z) is infinite of order [mu] 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 = [oo], where R(s, z) 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 [mu] where R (s, z) is infinite, and more generally equal to the sum of the values of m where R(s, z) = A; this we express by saying that a rational function R(s, z) takes any value (including [oo]) the same number of times on 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, z) = 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, z, which are connected by [f](s, z) = 0; about any place (c, d) for which z = c + P(t), s = d + Q(t), expand the product

dz R(s, z) -- dt

in powers of t and pick out the coefficient of t^(-1). There is only a finite number of places of this kind. The theorem is that the sum of these coefficients of t^(-1) is zero. This we express by _ _ | dz | |R(s, z) -- | = 0. |_ dt _|t^(-1)

The theorem holds for the case n = 1, that is, for rational functions of one variable z; in that case, about any finite point we have z - c = t, and about z = [oo] we have z^(-1) = t, and therefore dz/dt = -t^(-2); in that case, then, the theorem is that in any rational function of z,

_ / A1 A2 A_m \ \ ( ----- + --------- + ... + --------- ) + Pz^h + Qz^(h - 1) + ... + R, /_ \z - a (z - a)^2 (z - a)^m /

the sum [Sigma]A1 of the sum of the residues at the finite poles is equal to the coefficient of 1/z in the expansion, in ascending powers of 1/z, about z = [oo]; an obvious result. In general, if for a finite place of the algebraic construct associated with [f](s, z) = 0, whose neighbourhood is given by z = c + t^r, s = d + Q(t), there be a coefficient of t^(-1) in R(s, z)dz/dt, this will be r times the coefficient of t^(-r) in R(s, z) or R[d + Q(t), c + t^r], namely will be the coefficient of t^(-r) in the sum of the r series obtainable from R[d + Q(t), c + t^r] by replacing t by [omega]t, where [omega] is an rth root of unity; thus the sum of the coefficients of t^(-1) in R(s, z)dz/dt for all the places which arise for z = c, and the corresponding values of s, is equal to the coefficient of (z - c)^(-1) in R(s1, z) + R(s2z) + ... + R(s_n, z), where s1, ... s_n are the n values of s for a value of z near to z = c; this latter sum [Sigma] R(s_i, z) is, however, a rational function of z only. Similarly, near z = [oo], for a place given by z^(-1) = t^r, s = d + Q(t), or s^(-1) = Q(t), the coefficient of t^(-1) in R(s, z)dz/dt is equal to -r times the coefficient of t^r in R[d + Q(t), t^(-r)], that is equal to the negative coefficient of z^(-l) in the sum of the r series R[d + Q([omega]t), t^(-r)], so that, as before, the sum of the coefficients of t^(-1) in R(s, z)dz/dt at the various places which arise for z = [oo] is equal to the negative coefficient of z^(-1) in the same rational function of z, [Sigma] R(s_i, 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 z,

1 dR(s, z) ------- -------; R(s, z) dz

at a zero of R(s, z) near which R(s, z) = t^mH(t), we have

1 dR(s, z) dz d ------- ------- -- = -- {[lambda] [R(s, z)]} R(s, z) dz dt dt

where [lambda] denotes the generalized logarithmic function, that is equal to

mt^(-1) + power series in t;

similarly at a place for which R(s, z) = t^(-[mu]) K(t); the theorem _ _ | 1 dR(s, z) dz | | ------- -------- -- | t^(-1) = 0 |_ R(s, z) dz dt _|

thus gives [Sigma]m = [Sigma][mu], or, in words, the total number of zeros of R(s, z) on the algebraic construct is equal to the total number of its poles. The same is therefore true of the function R(s, z) - 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, z) = A on the algebraic construct.

We have seen above that all single valued doubly periodic meromorphic functions, with the same periods, are rational functions of two variables s, z connected by an equation of the form s^2 = 4z^3 + Az + B. Taking account of the relation connecting these variables s, z with the argument of the doubly periodic functions (which was above denoted by z), it can then easily be seen that the theorem now proved is a generalization of the theorem proved previously establishing for a doubly periodic function a definite _order_. There exists a generalization of another theorem also proved above for doubly periodic functions, namely, that the sum of the values of the argument in one parallelogram of periods for which a doubly periodic function takes a given value is independent of that value; this generalization, known as Abel's Theorem, is given S 17 below.

S 17. _Integrals of Algebraic Functions._--In treatises on Integral Calculus it is proved that if R(z) denote any rational function, an indefinite integral [int]R(z)dz can be evaluated in terms of rational and logarithmic functions, including the inverse trigonometrical functions. In generalization of this it was long ago discovered that if s^2 = az^2 + bz + c and R(s, z) be any rational function of s, z any integral [int]R(s, z)dz can be evaluated in terms of rational functions of s, z and logarithms of such functions; the simplest case is [int]s^(-1)dz or [int](az^2 + bz + c)^(-1/2)dz. More generally if f(s, z) = 0 be such a relation connecting s, z that when [theta] is an appropriate rational function of s and z both s and z are rationally expressible, in virtue of [f](s, z) = 0 in terms of [theta], the integral [int]R(s, z)dz is reducible to a form [int]H ([theta])d[theta], where H([theta]) is rational in [theta], and can therefore also be evaluated by rational functions and logarithms of rational functions of s and z. It was natural to inquire whether a similar theorem holds for integrals [int]R(s, z)dz wherein s^2 is a cubic polynomial in z. The answer is in the negative. For instance, no one of the three integrals

_ _ _ / dz / zdz / dz | --, | ---, | -------- _/ s _/ s _/ (z - c)s

can be expressed by rational and logarithms of rational functions of s and z; but it can be shown that every integral [int]R(s, z)dz can be expressed by means of integrals of these three types together with rational and logarithms of rational functions of s and z (see below under S 20, Elliptic Integrals). A similar theorem is true when s^2 = quartic polynomial in z; in fact when s^2 = A(z - a)(z - b)(z - c)(z - d), putting y = s(z - a)^(-2), x = (z - a)^(-1), we obtain y2 = cubic polynomial in x. Much less is the theorem true when the fundamental relation [f](s, z) = 0 is of more general type. There exists then, however, a very general theorem, known as _Abel's Theorem_, which may be enunciated as follows: Beside the rational function R(s, z) occurring in the integral [int]R(s, z)dz, consider another rational function H(s, z); let (a1), ... (a_m) denote the places of the construct associated with the fundamental equation [f](s, z) = 0, for which H(s, z) is equal to one value A, each taken with its proper multiplicity, and let (b1), ... (b_m) denote the places for which H(s, z) = B, where B is another value; then the sum of the m integrals [int] [(b_i) to (a_i)] R(s, z)dz is equal to the sum of the coefficients of t^(-1) in the expansions of the function

dz / H(s, z) - B \ R(s, z) -- [lambda] ( ----------- ), dt \ H(s, z) - A /

where [lambda] denotes the generalized logarithmic function, at the various places where the expansion of R(s, z)dz/dt contains negative powers of t. This fact may be obtained at once from the equation _ _ | 1 dz | | -------------- R(s, z) -- | = 0, |_ H(s, z) - [mu] dt _|t^(-1)

wherein [mu] is a constant. (For illustrations see below, under S 20, Elliptic Integrals.)

S 18. _Indeterminateness of Algebraic Integrals._--The theorem that the integral [int][a to x] [f](z)dz is independent of the path from a to z, holds only on the hypothesis that any two such paths are equivalent, that is, taken together from the complete boundary of a region of the plane within which [f](z) is finite and single valued, besides being differentiable. Suppose that these conditions fail only at a finite number of isolated points in the finite part of the plane. Then any path from a to z is equivalent, in the sense explained, to any other path together with closed ~~ paths beginning and ending at the arbitrary point a each enclosing one or more of the exceptional points, these closed paths being chosen, when [f](z) is not a single valued function, so that the final value of [f](z) at a is equal to its initial value. It is necessary for the statement that this condition may be capable of being satisfied.