Page:EB1911 - Volume 11.djvu/324

 writing ＝ƒ(x, y)＝ƒ(z − iy, y), the fact that this is really independent of y leads at once to ∂ƒ/∂x + i∂ƒ/∂y＝0, and hence to

so that is not any arbitrary function of x, y, and when  is known is determinate save for an additive constant. Also, in virtue of these equations, if, ′ be the values of corresponding to two near values of z, say z and z′, the ratio (′ − )/(z′ − z) has a definite limit when z′＝z, independent of the ultimate phase of z′ − z, this limit being therefore equal to ∂/∂x, that is, ∂/∂x + i∂/∂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 ∂/∂x, ∂/∂x are not zero, and P′, P″ be two points near to P on these curves respectively, and the corresponding points of the -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, 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 -plane with the same angles; the magnification, however, which is equal to $$\left [ \left ( \frac{\partial \xi}{\partial x}\right )^2 + \left ( \frac{\partial \xi}{\partial y}\right )^2 \right ]^{\frac{1}{2}} $$ 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 is a polynomial in z,

＝p0zn + p1zn−1 +. . . + pn&#8198;;

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 || > H; consider the lower limit of || for |z| < R; as 2 + 2 is a real continuous function of x, y for |z| < R, there is a point (x, y), say (x0, y0), at which || is least, say equal to, and therefore within a circle in the -plane whose centre is the origin, of radius , there are no points representing values corresponding to |z| < R. But if 0 be the value of  corresponding to (x0, y0), and the expression of − 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 /m + i sin /m), will correspond two points near to 0, say 1, and 20 − ′1, situated so that 0 is between them. One of these must be within the circle. We infer then that ＝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 −, where denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

§ 3. Limiting Operations.—In order that a complex number ＝ + i may have a limit it is necessary and sufficient that each of and  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, a number m exists such that for every n > m, and every positive p, the batch of terms wn+wn+1++wn+p is less than 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 and m, for all the points z of this range.

The infinite series of most importance are those of which the general term is anzn, wherein an 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, |anz0n| < 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&#8202;′ for which r&#8202;′ < |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 anzn, bnzn 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 an zn 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, |an| < Mr1−n. If in every arbitrarily small neighbourhood of z=0 there be a point for which two converging power series an zn, bnzn agree in value, then the series are identical, or an＝bn; thus also if anzn 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 ƒ(z)＝anzn of radius of convergence R, if |z0| < R and we put z＝z0 + t with |t| < R-|z0|, the resulting series an(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 ƒ(z)＝ Antn; hence A0＝ƒ(z0), and we have [ƒ(z0 + t) − ƒ(z0)]/t = n=1 Ant&#8202;n−1, 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 nanz0n−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 ƒ(z0+t)＝tnƒ(n)(z0)/n!, where ƒ(n)(z0) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about z＝0 for anzn, we infer that for the series ƒ(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 ƒ(z).

Perhaps the simplest possible power series is ez＝exp (z)＝1 + z2/2! + z3/3! + of which the radius of convergence is infinite. By multiplication we have exp (z).exp (z1)＝exp (z+z1). 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＝y&#8202;2 − 1 + cos y, U2＝y&#8202;3 − y + sin y, V2＝y&#8202;4 − y&#8202;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 − y&#8202;2 + y&#8202;4 −. . ., &emsp; sin y＝y − y&#8202;3 + y&#8202;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 + 2i)＝exp (x) [cos (y+2) + i sin(y+2)],

which we express by saying that exp (z) has the period 2i, and hence also the period 2ki, 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 and plotting the function ＝exp (z) upon a new plane, it follows at once from what has been said that every complex value of arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation ＝exp (z) thus defines z, regarded as depending upon, with only an additive ambiguity 2ki, where k is an integer. We write z＝; when is real this becomes the logarithm of ; in general ＝log || + i ph + 2ki, where k is an integer; and when describes a closed circuit surrounding the origin the phase of increases by 2, or k increases by unity. Differentiating the series for we have d/dz＝, so that z, regarded as depending upon, is also differentiable, with dz/d＝−1. On the other hand, consider the series − 1 − (−1)2 + (−1)3 − ; it converges when ＝2 and hence converges for ( − 1)2 −, that is, (1 + − 1)−1. Wherefore if denote this series, for | − 1| < 1, the difference −, regarded as a function of  and , has vanishing differential coefficients; if we take the value of  which vanishes when ＝1 we infer thence that for | − 1| < 1, ＝ n=1 $(−1)^{(n−1)}⁄n$(−1)n. It is to be remarked that it is impossible for while subject to | − 1| < 1 to make a circuit about the origin. For values of for which | − 1| ≮ 1, we can also calculate  with the help of infinite series, utilizing the fact that (′)＝) + ′).
 * − 1| < 1; its differential coefficient is, however, 1 − ( − 1) +

The function is required to define a when  and a are complex numbers; this is defined as exp [a], that is as n=0 an []n/n!. When a is a real integer the ambiguity of is immaterial here, since exp [a+2kai]＝exp [a]; when a is of the form 1/q, where q is a positive integer, there are q values possible for 1/q, of the form exp $$\left [ \frac{1}{q} \lambda (\xi ) \right ]$$ exp $$\left ( \frac{2k \pi i}{q} \right ) ,$$ 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 ; when a＝p/q, where p, q are integers without common factor, q being positive, we have p/q＝(1/q)p. The definition of the symbol 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&#8202;i; the number i is of modulus unity and phase ; thus (i)＝i (+2k); thus

i&#8202;i＝exp (− − 2k)＝exp (−) exp (−2k),

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