FUNCTIONS OF COMPLEX VARIABLES I. Complex Numbers and Their Geometric Representa tion.—The operations of addition, subtraction, multiplication and division with non-vanishing divisor, when applied to real numbers yieici only real numbers, but this does not hold good in the case of root extraction. We must, for example, either regard/ —1 as indicating an impossible operation, or else we must enlarge the number system so as to give a meaning to this symbol. The pur pose last indicated is accomplished by the definitions of complex numbers (see COMPLEX NUMBERS) and the elementary operations associated with them. It is only when complex numbers are ad mitted that we can assert the fundamental theorem of algebra according to which every algebraic equation with real coefficients has a solution.
A complex number is indicated by the symbol a+ib, where a and b are real numbers, and i is the so-called imaginary unit, sometimes written V/ — I since, by definition, —1. Addition, subtraction, multiplication and division are defined by the follow ing formulae, together with the assumption that the associative, commutative and distributive laws (see under those headings) hold for complex as well as for real numbers: (a-Fib)+(c-}-id) _ (a+ib) —(c+id) = (a—c)+i(b—d), (a+ib) (c+id) = (ac—bd)+i(ad+bc), a+ib (a+ib) (c —id) ac + i be — ad 2 + ) o c+id (c+id) (c —id) + d2 (c If b is zero, a+ib is identified with the real number a, so that the system of complex numbers includes that of real numbers. If b is not zero, a+ib is said to be imaginary and a is the real part of a+ib. The numbers a+ib and a—ib are then conjugate imagin aries; the sum and the product of two conjugate imaginaries are both real. Two complex numbers a+ib and c+id are equal when and only when a= c and b=d.
As real numbers may be represented by points on a straight line, so a one-to-one correspondence exists between all complex num bers (x, y) and the points of the plane whose coordinates are (x, y) in a given rectangular system (see ANALYTIC GEOMETRY). It is therefore customary to use the words point and number inter changeably for z = x+iy. The operations of adding and subtract ing two complex numbers corresponding to points and P2 give the points P3, P4 corresponding to a vector addition and subtrac tion, respectively, of directed segments from the origin of co ordinates to Pi and P2. (See VECTOR ANALYSIS.) A geometric picture of multiplication and division is most simply obtained by using a polar coordinate system in which the radius vector r = OP is always positive, and 0 is the angle from the positive x axis OX to OP. From elementary • trigonometry we have z = x+iy = r(cos e+i sin 0). The rule already adopted for multiplication gives, after a trigonometric reduction, sin sin = 02) Hence to obtain the radius vector, or modulus, and the angle of the product of two complex numbers, we multiply their moduli and add their angles. Similarly, in division we divide moduli and subtract angles. These definitions may be extended to products and quotients involving several terms. In particular, if m is a positive integer we have zm = rm (cos rn a +i sin nn e) , and in accordance with the usual definitions the same formula holds if m is a negative integer, or any rational number. Here, however, it must be observed that if is the angle of z that lies between o and 27 then for all integral values of n, positive or negative, is also an angle of z. If each of these possible values of 0 is used in the formula for zm where m is the reciprocal of an integer p, it will be seen that z'/P has p values.
From the geometric theorem that a straight line segment is shorter than any broken line joining its extremities we deduce the result that the modulus of a sum of complex numbers is not greater than the sum of the moduli of the terms, or, to use a customary notation, • • • +znl < . • • +IzfI.
Since the totality of complex numbers corresponds to that of all points of the plane, the domain of a complex variable is in general two-dimensional. A region is a domain consisting of all points of a single piece of the plane bounded by one or more closed curves; if its boundary consists of one closed curve of a single piece that does not cut itself, a region is said to be simply con nected, otherwise it is multiply connected. As to the boundary curves, we shall hereafter suppose them to be such as to allow the integration of a continuous function over them (see § 5, below). A region is open if it does not include its boundary; it is closed if all points of the boundary are included. A neighbourhood of a point is an open region to which the point belongs.
A function w = f (z) = u(x, y) +iv(x, y), defined so as to be single-valued at each point of a neighbourhood of z = Z = except possibly at Z itself, is said to approach the limit W = U-+iV as z approaches Z, provided we have limz_x, y) = U, , y) = V.
An equivalent definition is the following: f(z) = W if for every positive number E there exists a positive number b such that 11(z) — WI < E for all points z other than Z for which z —Z1 < b. The function f (z) is continuous at Z if lim,. z f (z) = f (Z) ; a necessary and sufficient condition is that both u(x, y) and v(x, y) be continuous at (X, Y). A function is continuous throughout a region if it is continuous at each point of the region.
3. Analytic Functions.—The definition of the derivative of a function of a complex variable is formally the same as for a real function of a real variable : it is clear that in terms of the two real variables Ax and Ay we are requiring the existence of a two-dimensional limit. Even when u and v have partial derivatives (i.e., derivatives with respect to one variable taken as though the other variable were constant) au/ax, au/ay, av/ax, av/ay, the derivative of f(z) may not exist. For example, if f (z) = x — iy, we have since Ay/A x has no limit that is independent of the way that and A x approach zero, the same is true of i f/Az. Functions of a complex variable that have derivatives thus form a restricted class, even among those that have derivatives with respect to x alone and with respect to y alone. A function single-valued in a region S is said to be analytic throughout that region if it has a derivative at each point of S; it is analytic at a point Z if there is a neighbourhood of Z throughout which it is analytic. Synonymous terms are monogenic, holomorphic, regular. It has been proved (by Ed. Goursat, 1900) that if f'(z) exists at each point of S, then f'(z) is a continuous function of z throughout S. Necessary and sufficient conditions that f'(z) exist are that the partial derivatives of u and v exist, are continuous and satisfy the Cauchy-Riemann equations, au/ax = av/ay, — av/ax.
In § 5 below it is shown that if f'(z) exists, then its derivative also exists at each point of S and similarly for the derived func tions of higher order. Thus the second partial derivatives of u and v also exist. By differentiating the first Cauchy-Riemann equation with respect to x and the second with respect to y, then Polynomials in z are analytic throughout the entire plane. Ra tional functions are analytic except at points where their denomin ators vanish. Exponential, logarithmic and trigonometric functions of z require definition. Thus we wish to define ez so that it will be single-valued and analytic throughout the entire plane and will be identical with the real exponential function e z when z = x; similarly for sin z and cos z. It can be shown that the only func tions satisfying these requirements are the following, sin cos they also satisfy the formal identities that hold for real expo nential and trigonometric functions. The logarithm of z (to the natural base e) is the function w= log z which identically satis fies the equation e'° = z. If (r, 0) are polar coordinates of z, we have log z = log r-Fi0, where logr is the real logarithm of the posi tive real number r. The angle 0 is, however, as we have seen, infinitely many-valued. Hence log z is infinitely many-valued ; any two of its values differ by an integral multiple of The inverse trigonometric functions can be expressed in terms of logarithms. From these elementary functions one may pass to algebraic functions and the elliptic and abelian functions asso ciated with them. (See ELLIPTIC FUNCTIONS, and the authori ties cited at the end.) 4. Geometrical and Physical Applications.—As has al ready been indicated, there were reasons of a purely mathematical nature for the introduction of complex numbers and functions of complex variables. Without them certain inverse operations on real numbers would have no meaning and many of the theorems of analysis would have awkward restrictions. In the applications of the theory of analytic functions in geometry and physics, how ever, is to be found one of the chief reasons why this theory has dominated analysis for the last three-quarters of a century. The Cauchy-Riemann equations and Laplace's equation are of central importance in the theory of maps and in various problems of mathematical physics (see SPHERICAL HARMONICS and CONFORMAL REPRESENTATION). Pairs of solutions, u and v, of those equa tions combine to form analytic functions u-Fiv of the complex variable x+iy. Thus theorems about analytic functions have in terpretations in such fields as those of conformal mapping and of the conduction of heat and electricity.
A conformal map may, for example, be described mathe matically as follows: A plane region S is said to be mapped upon another plane region in a one-to-one manner if to each point of one region corresponds one and only one point of the other. If a rectangular co-ordinate system (u, v) is set up for and a system (x, y) for S, it follows that u and v are functions of x and y; if these functions are continuous, S is said to be mapped continuously on Z. To a piece of curve of length As in S will correspond a piece of length Aa in Z. If As approaches zero, while one end-point is kept fixed, the ratio of 0 r to As may approach no limit, or it may approach different limits on different curves, or it may approach the same limit on every curve through the fixed point. In the last case this limit is the scale of the map at that point. If a continuous one-to-one map has a scale at every point it is said to be conformal, and it can be shown that if two curves intersect at a point of S the corre sponding curves intersect at the same angle in X . It can then be proved that either the pair (u, v), or the pair (u, —v) satisfies the Cauchy-Riemann equations. Every analytic function w= f(z) has, as we have seen, a real part u and a pure imaginary part iv such that u and v satisfy the Cauchy-Riemann equations. It therefore defines a conformal map of a suitably chosen z region on a w-region. The correspondence of two geographic maps, such as a stereographic map and a Mercator's map of the same portion of the earth's surface, is completely described by an appropriate equation w= f(z). By a theorem of Riemann's, any simply connected plane region whatever can be mapped on the interior of a circle of unit radius about the origin by a suitable analytic function. In particular, every region, finite or infinite, bounded by a single circle or straight line is so mapped by a suitable linear transformation w= (az+b)/(cz+d).
It would be impossible to discuss here the many physical appli cations of the theory of analytic functions; a brief reference to the steady flow of heat in a conducting plate must suffice. Sup pose a thin homogeneous metal plate is insulated except at its edge. Each point of the edge is kept at a fixed temperature, but this may vary from point to point. After a time each point (x, y) of the plate reaches, and maintains thereafter, a temperature u(x, y) ; the flow of heat is then said to be steady. It can be shown that u satisfies Laplace's equation. Curves along which u is constant are called isotherms, while lines of flow are the curves v = constant that cut each isotherm perpendicularly. The fact that u and v must satisfy the Cauchy-Riemann equations again makes contact with the theory of analytic functions.
5. Integrals.—In order to obtain a definition of integration formally similar to that used for a real function of a real variable, we consider a curve segment C interior to a region S throughout which f(z) is analytic. On C we take n+ i successive points zo, zi, z2, . . . , z„, of which the first, and the last, z„ = Z are fixed. If the sum has a limit L as we take successive subdivision schemes, such that the limit of the largest modulus I — is zero, and if L is the same for all such sequences of sums formed for the function f(z) and for the curve C from to Z, then this limit is said to be the value of the definite integral of f(z) on C from to Z, and is written (z)dz. In terms of the real functions u(x, y), v(x, y), where f(z) = u+iv, this definition yields the identity f (z)dz udx —vdy+i vdx+udy, where the integrals on the right are real curve or line integrals along C (see CALCULUS). If f(z) is continuous on C, and if C is of finite length, or in particular if C is regular (composed of a finite number of pieces with continuously turning tangent), the integral exists.
If f(z) is merely continuous throughout a region S, the value of the integral of f(z) from to Z will depend, in general, on the curve of integration. But if f(z) is analytic throughout a simply connected region S, then the integral of f (z) from to Z is the same on all regular curves in S that join and Z. This follows from Cauchy's integral theorem (1825; proved by E. Goursat, 1900, without assuming that f(z) is continuous), which states that if f(z) is continuous in a closed region S, simply or multiply connected, and analytic throughout open S, then the integral of f(z) around the whole boundary of S is zero, provided each boundary curve is so traversed as to leave the interior of S to the left. Hence if curves and C2 within S extend from to Z and bound a simply connected portion S' of S, then the integral of f(z) around the boundary of S' is zero; that is, the forward inte gral on plus the backward integral on C2 vanishes, so that the forward integral on C1 is equal to the forward integral on C2. The same result is true even when C1 and C2 intersect at points be tween and Z, provided S is simply connected. In this case, then, F(z) =f f(z)dz has a well-defined meaning. It is easily 20 proved that f(z) is the derivative of F(z). If f(z) is analytic throughout S except at a single interior point then the integral of f(z) around the boundary C is called the residue of f(z) at Here C can be replaced by a circle of small radius about On these considerations Cauchy built his calculus of residues, one aim of which was to obtain evaluations of improper real integrals.
A most important consequence of Cauchy's integral theorem is the integral formula, which holds if f(t) is single-valued and analytic on the open region S bounded by C, and continuous on closed S, while z is an interior point of S. Thus the function f(z) is uniquely determined at each point of S by its boundary values. Since the above inte gral can be differentiated with respect to z by differentiating the integrand, it follows that an analytic function has derivatives of all orders, the nth derivative being given by the formula Among the corollaries of the integral formula are the following: (1) f(z) has no maximum in the interior of S; (2) if S consists of the entire plane, then either f(z) is not everywhere finite (in which case it is called an integral function), or else it is a con stant; (3) every algebraic equation f(z) =o, where f(z) is a polynomial of degree greater than zero, has at least one solution (the so-called fundamental theorem of algebra).
A series of functions ... +w„(z)+ .. • may converge in some regions and diverge in others. Such a series is uniformly convergent throughout a region S if for every positive number e there exists an integer m such that I sm(z) (z) I < e for all values of n greater than m; here designates the sum of the first n terms of the series and the same m must apply for all points z of S. If each of the terms of a series is single-valued and analytic and the series is uniformly convergent in 5, then the sum-function is analytic in S.
Among the most important series are the power series zo) ... +cm(z—zo)"+ .... Such a series may converge for all values of z, or for z = only; other wise there is a circle of convergence with its centre at zo, through out the interior of which the series converges and outside which it diverges. Such a series is uniformly convergent within every region interior to its circle of convergence, and hence represents an analytic function throughout the open region interior to the circle. Conversely, every function f(z) analytic throughout the interior of a circle is there represented by a power series, called its Taylor's series. For the coefficients we have the formula ck= f k Such a series may be differentiated and integrated term by term ; the new series will represent the derivative and the integral of the sum of the original series throughout its circle of convergence. Since the values of the derivatives of f(z) at zo determine its Taylor's series, it follows that these values com pletely characterize the function itself throughout the circle of convergence. A more general result may be stated as follows: If f (z) is single-valued and analytic throughout S it is completely determined in that region by its value and those of all its deriva tives at an interior point, or by its values at the points of an infinite set having a limiting point within S.
A series in which both positive and negative powers of z—zo occur is called a Laurent's series. Such a series has a ring of con vergence composed of the region bounded by two concentric circles about A function analytic and single-valued in such a ring is representable there by a Laurent's series. Infinite products of analytic functions have also been the subject of many investi gations.
The definition and classification of analytic functions in terms of their singularities was the starting-point of the program of Riemann for the theory of functions of a complex variable. As an example, we may define a rational function (with suitable con ventions regarding z=) as one with no singularities throughout the plane, or at z = oo , except poles.
8. Analytic Continuation: Many-valued Functions.—If f(z) is analytic in S and 4, (z) analytic in S'; if, further, S and S' overlap and f (z) = 4)(z) throughout this overlapping portion, then 4)(z) is uniquely determined and is said to be an analytic continuation of f(z) into S'. We may thus be able to continue a function analytically by a chain of overlapping regions ; the totality of functional values defined when this process has been carried as far as it will go is called a monogenic analytic function. In particular such a monogenic function may be many-valued when a chain of regions used in its definition overlaps itself. We then think of the plane of z as replaced by a many-sheeted sur face called a Riemann surface, and of the many-valued monogenic function as single-valued on its Riemann surface. This concep tion has been especially fruitful in the study of algebraic func tions and their integrals.
9. Functions of Two or More Complex Variables.—Many of the formulae for functions of one complex variable are readily extended to functions of more than one such variable. In other directions new difficulties appear. Algebraic functions of several variables have received especial attention, but the general field of functions of more than one variable is comparatively undeveloped as yet.
For the Functions of Complex Variables see Ency. der Math. Wiss., mentioned above. The report of A. Brill and M. Nother, "Die Entwickelung der Theorie der allg. Funkt. in alterer and neuerer Zeit." Jahresber. d. Deutsch. Math. Ver., 3 (1894), gives an excellent historical account. General treatises carrying the subject into the present century are: W. F. Osgood, Lehrbuch der Funktionentheorie (Leipzig, Band i., 4th ed., 1923, Band ii., 1927-28) ; L. Bieberbach, Lehrb. d. Funktionentheorie (Leipzig, 1923, 1927) ; A. Hurwitz and R. Courant, Vorles. ii. aIlg. Funktionentheorie a. elliptische Funkt. (2d ed., 1925) ; K. Hensel and G. Landsberg, Theorie d. algebr. Funkt. einer Variablen, etc. (Leipzig, 1902) ; H. Burkhardt, Einfiihrung in die Theorie d. analyt. Funkt. einer komplexen Veranderlichen (3rd ed., Leipzig, 1908) , translated into English by S. E. Rasor (1913) ; J. Pierpont, Functions of a Complex Variable (1914) ; A. R. Forsyth, Theory of Functions of a Complex Variable (3d ed., 1918), and Lectures Introductory to the Theory of Functions of Two Complex Variables (1914) ; E. Goursat, Cours d'Analyse, vol. ii. (4th ed., 1925), translated into English by E. R. Hedrick and O. Dunkel (1916) ; E. Picard, Traite d'Analyse, vol. ii. (3d ed., 1926) .
The series of monographs on the Theory of Functions edited by E. Borel (Paris, 1898— ) also contains many important contributions.
(D. R. C.)