Part 24

In the preceding section the doctrine of functionality is discussed with respect to real quantities; in this section the theory when complex or imaginary quantities are involved receives treatment. The following abstract explains the arrangement of the subject matter: (S 1), _Complex numbers_, states what a complex number is; (S 2), _Plotting of simple expressions involving complex numbers_, illustrates the meaning in some simple cases, introducing the notion of conformal representation and proving that an algebraic equation has complex, if not real, roots; (S 3), _Limiting operations_, defines certain simple functions of a complex variable which are obtained by passing to a limit, in particular the exponential function, and the generalized logarithm, here denoted by [lambda](z); (S 4), _Functions of a complex variable in general_, after explaining briefly what is to be understood by a region of the complex plane and by a path, and expounding a logical principle of some importance, gives the accepted definition of a function of a complex variable, establishes the existence of a complex integral, and proves Cauchy's theorem relating thereto; (S 5), _Applications_, considers the differentiation and integration of series of functions of a complex variable, proves Laurent's theorem, and establishes the expansion of a function of a complex variable as a power series, leading, in (S 6), _Singular points_, to a definition of the region of existence and singular points of a function of a complex variable, and thence, in (S 7), _Monogenic Functions_, to what the writer believes to be the simplest definition of a function of a complex variable, that of Weierstrass; (S 8), _Some elementary properties of single valued functions_, first discusses the meaning of a pole, proves that a single valued function with only poles is rational, gives Mittag-Leffler's theorem, and Weierstrass's theorem for the primary factors of an integral function, stating generalized forms for these, leading to the theorem of (S 9), _The construction of a monogenic function with a given region of existence_, with which is connected (S10), _Expression of a monogenic function by rational functions in a given region_, of which the method is applied in (S 11), _Expression of_ (1 - z)^(-1) _by polynomials_, to a definite example, used here to obtain (S 12), _An expansion of an arbitrary function by means of a series of polynomials, over a star region_, also obtained in the original manner of Mittag-Leffler; (S 13), _Application of Cauchy's theorem to the determination of definite integrals_, gives two examples of this method; (S 14), _Doubly Periodic Functions_, is introduced at this stage as furnishing an excellent example of the preceding principles. The reader who wishes to approach the matter from the point of view of Integral Calculus should first consult the section (S 20) below, dealing with _Elliptic Integrals_; (S 15), _Potential Functions, Conformal representation in general_, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (S 16), _Multiple-valued Functions, Algebraic Functions_, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (S 17), _Integrals of Algebraic Functions_, enunciating Abel's theorem; (S 18), _Indeterminateness of Algebraic Integrals_, deals with the periods associated with an algebraic integral, establishing that for an elliptic integral the number of these is two; (S 19), _Reversion of an algebraic integral_, mentions a problem considered below in detail for an elliptic integral; (S 20), _Elliptic Integrals_, considers the algebraic reduction of any elliptic integral to one of three standard forms, and proves that the function obtained by reversion is single-valued; (S 21), _Modular Functions_, gives a statement of some of the more elementary properties of some functions of great importance, with a definition of Automorphic Functions, and a hint of the connexion with the theory of linear differential equations; (S 22), _A property of integral functions, deduced from the theory of modular functions_, proves that there cannot be more than one value not assumed by an integral function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the periods; (S 23), _Geometrical applications of Elliptic Functions_, shows that any plane curve of deficiency unity can be expressed by elliptic functions, and gives a geometrical proof of the addition theorem for the function RN(u); (S 24), _Integrals of Algebraic Functions in connexion with the theory of plane curves_, discusses the generalization to curves of any deficiency; (S 25), _Monogenic Functions of several independent variables_, describes briefly the beginnings of this theory, with a mention of some fundamental theorems: (S 26), _Multiply-Periodic Functions and the Theory of Surfaces_, attempts to show the nature of some problems now being actively pursued.

Beside the brevity necessarily attaching to the account here given of advanced parts of the subject, some of the more elementary results are stated only, without proof, as, for instance: the monogeneity of an algebraic function, no reference being made, moreover, to the cases of differential equations whose integrals are monogenic; that a function possessing an algebraic addition theorem is necessarily an elliptic function (or a particular case of such); that any area can be conformally represented on a half plane, a theorem requiring further much more detailed consideration of the meaning of _area_ than we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.

S 1. _Complex Numbers._--Complex numbers are numbers of the form x + iy, where x, y are ordinary real numbers, and i is a symbol imagined capable of combination with itself and the ordinary real numbers, by way of addition, subtraction, multiplication and division, according to the ordinary commutative, associative and distributive laws; the symbol i is further such that i^2 = -1.

Taking in a plane two rectangular axes Ox, Oy, we assume that every point of the plane is definitely associated with two real numbers x, y (its co-ordinates) and conversely; thus any point of the plane is associated with a single complex number; in particular, for every point of the axis Ox, for which y = O, the associated number is an ordinary real number; the complex numbers thus include the real numbers. The axis Ox is often called the real axis, and the axis Oy the imaginary axis. If P be the point associated with the complex variable z = x + iy, the distance OP be called r, and the positive angle less than 2[pi] between Ox and OP be called [theta], we may write z = r(cos[theta] + i sin[theta]); then r is called the modulus or absolute value of z and often denoted by |z| and [theta] is called the phase or amplitude of z, and often denoted by ph (z); strictly the phase is ambiguous by additive multiples of 2[pi]. If z' = x' + iy' be represented by P', the complex argument z' + z is represented by a point P" obtained by drawing from P' a line equal to and parallel to OP; the geometrical representation involves for its validity certain properties of the plane; as, for instance, the equation z' + z = z + z' involves the possibility of constructing a parallelogram (with OP" as diagonal). It is important constantly to bear in mind, what is capable of easy algebraic proof (and geometrically is Euclid's proposition III. 7), that the modulus of a sum or difference of two complex numbers is generally less than (and is never greater than) the sum of their moduli, and is greater than (or equal to) the difference of their moduli; the former statement thus holds for the sum of any number of complex numbers. We shall write E(i[theta]) for cos[theta] + i sin [theta]; it is at once verified that E(i[alpha]). E(i[beta]) = E[i([alpha] + [beta])], so that the phase of a product of complex quantities is obtained by addition of their respective phases.

S 2. _Plotting and Properties of Simple Expressions involving a Complex Number._--If we put [zeta] = (z-i)/(z + i), and, putting [zeta] = [xi] + i[eta], take a new plane upon which [xi], [eta] are rectangular co-ordinates, the equations [xi] = (x^2 + y^2-1)/[x^2 + (y + 1)^2], [eta] = -2xy/[x^2 + (y + i)^2] will determine, corresponding to any point of the first plane, a point of the second plane. There is the one exception of z = -i, that is, x = 0, y = -1, of which the corresponding point is at infinity. It can now be easily proved that as z describes the real axis in its plane the point [zeta] describes once a circle of radius unity, with centre at [zeta] = 0, and that there is a definite correspondence of point to point between points in the z-plane which are above the real axis and points of the [zeta]-plane which are interior to this circle; in particular z = i corresponds to [zeta] = 0.

Moreover, [zeta] being a rational function of z, both [xi] and [eta] are continuous differentiable functions of x and y, save when [zeta] is infinite; writing [zeta] = [f](x, y) = [f](z - iy, y), the fact that this is really independent of y leads at once to (Pd)f/(Pd)x + i(Pd)[f]/(Pd)y = 0, and hence to

(Pd)[xi] (Pd)[eta] (Pd)[xi] (Pd)[eta] (Pd)^2[xi] (Pd)^2[xi] -------- = ---------, -------- = - ---------, ---------- + ---------- = 0; (Pd)x (Pd)y' (Pd)y (Pd)x' (Pd)x^2 (Pd)y^2

so that [xi] is not any arbitrary function of x, y, and when [xi] is known [eta] is determinate save for an additive constant. Also, in virtue of these equations, if [zeta], [zeta]' be the values of [zeta] corresponding to two near values of z, say z and z', the ratio ([zeta]'-[zeta])/(z'- z) has a definite limit when z' = z, independent of the ultimate phase of z'- z, this limit being therefore equal to (Pd)[zeta]/(Pd)x, that is, (Pd)[xi]/(Pd)x + i(Pd)[eta])/(Pd)x. Geometrically this fact is interpreted by saying that if two curves in the z-plane intersect at a point P, at which both the differential coefficients (Pd)[xi]/(Pd)x, (Pd)[eta]/(Pd)x are not zero, and P', P" be two points near to P on these curves respectively, and the corresponding points of the [zeta]-plane be Q, Q', Q", then (1) the ratios PP"/PP', QQ"/QQ' are ultimately equal, (2) the angle P'PP" is equal to Q'QQ", (3) the rotation from PP' to PP" is in the same sense as from QQ' to QQ", it being understood that the axes of [xi], [eta] in the one plane are related as are the axes of x, y. Thus any diagram of the z-plane becomes a diagram of the [zeta]-plane with the same angles; the magnification, however, which is equal to _ _ | /(Pd)[xi]\^2 /(Pd)[xi]\^2 | 1/2 | ( -------- ) + ( -------- ) | |_ \ (Pd)x / \ (Pd)y / _|

varies from point to point. Conversely, it appears subsequently that the expression of any copy of a diagram (say, a map) which preserves angles requires the intervention of the complex variable.

As another illustration consider the case when [zeta] is a polynomial in z,

[zeta] = p0 z^n + p1 z^(n - 1) + ... + p_n;

H being an arbitrary real positive number, it can be shown that a radius R can be found such for every |z| > R we have |[zeta]| > H; consider the lower limit of |[zeta]| for |z| < R; as [xi]^2 + [eta]^2 is a real continuous function of x, y for |z| < R, there is a point (x, y), say (x0, y0), at which |[zeta]| is least, say equal to [rho], and therefore within a circle in the [zeta]-plane whose centre is the origin, of radius [rho], there are no points [zeta] representing values corresponding to |z| < R. But if [zeta]0 be the value of [zeta] corresponding to (x0, y0), and the expression of [zeta] - [zeta]0 near z0 = x0 + iy0, in terms of z - z0, be A(z - z0)^m + B(z - z0)^(m + 1) + ..., where A is not zero, to two points near to (x0, y0), say (x1, y1) or z1 and z2 = z0 + (z1 - z0)(cos [pi]/m + i sin [pi]/m), will correspond two points near to [zeta]0, say [zeta]1, and 2[zeta]0 -[zeta]'1, situated so that [zeta]0 is between them. One of these must be within the circle ([rho]). We infer then that [rho] = 0, and have proved that every polynomial in z vanishes for some value of z, and can therefore be written as a product of factors of the form z - [alpha], where [alpha] denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

S 3. _Limiting Operations._--In order that a complex number [zeta] = [xi] + i[eta] may have a limit it is necessary and sufficient that each of [xi] and [eta] has a limit. Thus an infinite series w0 + w1 + w2 + ..., whose terms are complex numbers, is convergent if the real series formed by taking the real parts of its terms and that formed by the imaginary terms are both convergent. The series is also convergent if the real series formed by the moduli of its terms is convergent; in that case the series is said to be absolutely convergent, and it can be shown that its sum is unaltered by taking the terms in any other order. Generally the necessary and sufficient condition of convergence is that, for a given real positive [epsilon], a number m exists such that for every n > m, and every positive p, the batch of terms w_n + w_(n + 1) + ... + w_(n + p) is less than [epsilon] in absolute value. If the terms depend upon a complex variable z, the convergence is called _uniform_ for a range of values of z, when the inequality holds, for the same [epsilon] and m, for all the points z of this range.

The infinite series of most importance are those of which the general term is a_nz^n, wherein a_n is a constant, and z is regarded as variable, n = 0, 1, 2, 3, ... Such a series is called a power series, if a real and positive number M exists such that for z = z0 and every n, |a_n z0^n| < M, a condition which is satisfied, for instance, if the series converges for z = z0, then it is at once proved that the series converges absolutely for every z for which |z| < |z0|, and converges uniformly over every range |z| < r' for which r' < |z0|. To every power series there belongs then a circle of convergence within which it converges absolutely and uniformly; the function of z represented by it is thus continuous within the circle (this being the result of a general property of uniformly convergent series of continuous functions); the sum for an interior point z is, however, continuous with the sum for a point z0 on the circumference, as z approaches to z0 provided the series converges for z = z0, as can be shown without much difficulty. Within a common circle of convergence two power series [Sigma] a_n z^n, [Sigma] b_n z^n can be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If r1 be less than the radius of convergence of a series [Sigma] a_nz^n and for |z| = r1, the sum of the series be in absolute value less than a real positive quantity M, it can be shown that for |z| = r1 every term is also less than M in absolute value, namely, |a_n| < Mr1^(-n). If in every arbitrarily small neighbourhood of z = 0 there be a point for which two converging power series [Sigma]a_nz^n, [Sigma]b_n z^n agree in value, then the series are identical, or a_n = b_n; thus also if [Sigma]a_nz^n vanish at z = 0 there is a circle of finite radius about z = 0 as centre within which no other points are found for which the sum of the series is zero. Considering a power series [f](z) = [Sigma]a_nz^n of radius of convergence R, if |z0| < R and we put z = z0 + t with |t| < R-|z0|, the resulting series [Sigma]a_n (z0 + t)^n may be regarded as a double series in z0 and t, which, since |z0| + t < R, is absolutely convergent; it may then be arranged according to powers of t. Thus we may write [f](z) = [Sigma]A_n t^n; hence A0 = [f](z0), and we have [[f](z0 + t) - [f](z0)]/t = [Sigma](n=1) A_n t^(n-l), wherein the continuous series on the right reduces to A1 for t = 0; thus the ratio on the left has a definite limit when t = 0, equal namely to A1 or [Sigma]na_nz0^(n - 1). In other words, the original series may legitimately be differentiated at any interior point z0 of its circle of convergence. Repeating this process we find [f](z0 + t) = [Sigma]t^n [f]^(n) (z0)/n!, where [f]^(n) (z0) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about z = 0 for [Sigma]a_n z^n, we infer that for the series [f](z) every point which reduces it to zero is an isolated point, and of such points only a finite number lie within a circle which is within the circle of convergence of [f](z).

Perhaps the simplest possible power series is e^z = exp(z) = 1 + z^2/2! + z^3/3! + ... of which the radius of convergence is infinite. By multiplication we have exp(z).exp(z^1) = exp(z + z^1). In

## particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy).

Now the functions

U0 = sin y, V0 = 1 - cos y, U1 = y - sin y, V1 = 1/2y^2 - 1 + cos y, U2 = (1/6)y^3 - y + sin y, V2 = (1/24)y^4 - 1/2y^2 + 1 - cos y, ...

all vanish for y = 0, and the differential coefficient of any one after the first is the preceding one; as a function (of a real variable) is increasing when its differential coefficient is positive, we infer, for y positive, that each of these functions is positive; proceeding to a limit we hence infer that

cos y = 1 - 1/2y^2 + (1/24)y^4 - ..., sin y = y - (1/6)y^3 + (1/120)y^5 - ...,

for positive, and hence, for all values of y. We thus have exp(iy) = cos y + i sin y, and exp (z) = exp (x).(cos y + i sin y). In other words, the modulus of exp (z) is exp (x) and the phase is y. Hence also

exp(z + 2[pi]i) = exp(x) [cos (y + 2[pi]) + i sin(y + 2[pi])],

which we express by saying that exp (z) has the period 2[pi]i, and hence also the period 2k[pi]i, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x = 0, we at once prove that exp (z) has no other periods.

Taking in the plane of z an infinite strip lying between the lines y = 0, y = 2[pi] and plotting the function [zeta] = exp (z) upon a new plane, it follows at once from what has been said that every complex value of [zeta] arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation [zeta] = exp(z) thus defines z, regarded as depending upon [zeta], with only an additive ambiguity 2k[pi]i, where k is an integer. We write z = [lambda]([zeta]); when [zeta] is real this becomes the logarithm of [zeta]; in general [lambda]([zeta]) = log |[zeta]| + i ph ([zeta]) + 2k[pi]i, where k is an integer; and when [zeta] describes a closed circuit surrounding the origin the phase of [zeta] increases by 2[pi], or k increases by unity. Differentiating the series for [zeta] we have d[zeta]/dz = [zeta], so that z, regarded as depending upon [zeta], is also differentiable, with dz/d[zeta] = [zeta]^(-1). On the other hand, consider the series [zeta] - 1 - 1/2([zeta] - 1)^2 + 1/3([zeta] - 1)^3 - ...; it converges when [zeta] = 2 and hence converges for |[zeta] - 1| < 1; its differential coefficient is, however, 1 - ([zeta] - 1) + ([zeta] - 1)^2 - ..., that is, (1 + [zeta] - 1)^(-1). Wherefore if [phi]([zeta]) denote this series, for |[zeta] - 1| < 1, the difference [lambda]([zeta]) - [phi]([zeta]), regarded as a function of [xi] and [eta], has vanishing differential coefficients; if we take the value of [lambda]([zeta]) which vanishes when [zeta] = 1 we infer thence that for |[zeta] - 1| < 1, [lambda]([zeta]) = [Sigma][n = 1] [(-1)^(n - 1)]/n ([zeta] - 1)^n. It is to be remarked that it is impossible for [zeta] while subject to |[zeta] - 1| < 1 to make a circuit about the origin. For values of [zeta] for which |[zeta] - 1| [not less than] 1, we can also calculate [lambda]([zeta]) with the help of infinite series, utilizing the fact that [lambda]([zeta][zeta]') = [lambda]([zeta]) + [lambda]([zeta]').

The function [lambda]([zeta]) is required to define [zeta]^a when [zeta] and a are complex numbers; this is defined as exp [a[lambda]([zeta])], that is as [Sigma] (n=0) a^n[[lambda] ([zeta])]^n/n!. When a is a real integer the ambiguity of [lambda]([zeta]) is immaterial here, since exp [a[lambda]([zeta]) + 2ka[pi]i] = exp[a[lambda]([zeta])]; when a is of the form 1/q, where q is a positive integer, there are q values possible for [zeta]^(1/q), of the form exp [1/q [lambda]([zeta])] exp(2k[pi]i/q), with k = 0, 1, ... q - 1, all other values of k leading to one of these; the qth power of any one of these values is [zeta]; when a = p/q, where p, q are integers without common factor, q being positive, we have [zeta]^(p/q) = ([zeta]^(1/q))^p. The definition of the symbol [zeta]^a is thus a generalization of the ordinary definition of a power, when the numbers are real. As an example, let it be required to find the meaning of i^i; the number i is of modulus unity and phase 1/2[pi]; thus [lambda](i) = i(1/2[pi] + 2k[pi]); thus

i^i = exp(-1/2[pi] - 2k[pi]) = exp(-1/2[pi]) exp(-2k[pi]),

is always real, but has an infinite number of values.

The function exp (z) is used also to define a generalized form of the cosine and sine functions when z is complex; we write, namely, cos z = 1/2[exp(iz) + exp(-iz)] and sin z = -1/2i[exp(iz) - exp(-iz)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity.

S4. _Of Functions of a Complex Variable in General._--We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in z. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation.