COMPLEX VARIABLE, Theory of Functions of a. 1. Function.— The Theory of Functions of a Complex Variable deals with ordinary complex numbers, ^=x yi, when x and y are real, and i denotes-the pure imaginary number Just as real numbers are repre sented by points on a line,• e.g., the points of the axis of x in analytic geometry, so complex numbers are represented by points in a plane, called the complex plane, z corresponding to that point whose Cartesian co-ordinates are (x, y). The distance V + of z from the point a=0 is called the absolute value of and is written 121. Let S denote a region of the plane bounded by one or more closed curves. To every point a of S let one or more numbers w, real or complex, be assigned according to some definite law. Then w is said to be a single- or multiple-valued function of a: tv=-1(2). For the present we shall restrict ourselves to single-valued functions.
2. Continuity and Limits.— The function to'=' f(z) is continuous in the region S if, when descnnes any continuous curve in S, the image w of a describes a continuous curve in the w plane. The variable f(z) is said to approach b as its limit, if, when a approaches a along an arbitrary path, the corresponding point so=f (a) approaches the point w=b; in symbols, lim f (a) = b.
C=IS A complex variable a is said to become infinite: , if its absolute value becomes infinite; i.e., if the corresponding point of the complex plane recedes indefinitely, no matter in what direction.
3. Derivative.—The function w=f(z) has a• derivative, f(z), if the quotient .1w/dz ap proaches a limit, when dz approaches 0: 431 lim fix dst —f(s)_ f'(z) D sw. ds =o When z is the independent variable, dw= Dzwd and it is shown, as in the differential calculus, that dz=dz and dw= D dz, the latter relation holding no matter what the independent variable may be. The general rules for differentiation apply here, e.g., dV, etc.
Moreover, we have n = pos. integer. ds Hence it follows that all rational integral func tions, i.e., polynomials in a, and all other ra
tional functions of a, can be differentiated ac cording to the same rules as reals.
Theorem: The necessary and sufficient con dition that the function w= u vi=f(x yi) have a continuous derivative is that the real functions u and v of the real variables x and y have continuous first partial derivatives satisfy ing the Cauchy-Riemann Differential Equations: au av auav _ _ . ( ) 1 ax ay ay az The condition is necessary. For since we shall always get one and the same limit, no mat ter along what path may approach 0, let dz pass first only through real values: dr = dx; thus lim dw ow Secondly, let re be pure imaginary: kg =i4Y: then lim dw I aw LW.
saY $ 0Y Hence OwI aw 7 and it remains only to separate the real and the pure imaginary terms. We omit the proof of the sufficiency of the condition.
Two functions which satisfy relations (1) are called conjugate functions.
Example: Let la = ez(cos y + i sin y). Then= ex— = cos y = , 8x ay au av —=—ez sin y ax 8y Hence w has a derivative, and furthermore dip A function which is single-valued through out a region S and has a continuous derivative in S is said to be analytic in S. The terms holomorphic, monogenic and synectic are also sometimes used in this sense.
4. Conformal Mapping.— Given any two functions u—.0(x,y), (2)we may interpret them geometrically as trans forming the points of one plane into the points (u, v) of a second plane. Thus a region S of the first plane will be mapped, if certain further conditions are fulfilled, in a one-to-one manner and continuously on a region E of the (u, v) plane. For example, let 114..= Here the first quadrant of the (x, y)-plane is mapped on the upper half of the (u, v)-plane, the family of lines u=const. going over into the family of equilateral hyperbolas whose axes lie in the co-ordinate axes, and the family of lines In 00?0* ?????.?? ?E?????? • 1111 ....1111 war •• •bid