Page:EB1911 - Volume 11.djvu/324

writing ζ＝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

∂ξ	＝	∂η	,	∂ξ	＝−	∂η	,	∂2ξ	+	∂2ξ	＝0;
∂x		∂y		∂y		∂x		∂x2		∂y2

so that ξ is not any arbitrary function of x, y, and when ξ is known η is determinate save for an additive constant. Also, in virtue of these equations, if ζ, ζ′ be the values of ζ corresponding to two near values of z, say z and z′, the ratio (ζ′ − ζ)/(z′ − z) has a definite limit when z′＝z, independent of the ultimate phase of z′ − z, this limit being therefore equal to ∂ζ/∂x, that is, ∂ξ/∂x + i∂η/∂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 ∂ξ/∂x, ∂η/∂x are not zero, and P′, P″ be two points near to P on these curves respectively, and the corresponding points of the ζ-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 ξ, η 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 ζ-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 ζ is a polynomial in z,

ζ＝p₀zⁿ + p₁zⁿ⁻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 |ζ| > H; consider the lower limit of |ζ| for |z| < R; as ξ² + η² is a real continuous function of x, y for |z| < R, there is a point (x, y), say (x₀, y₀), at which |ζ| is least, say equal to ρ, and therefore within a circle in the ζ-plane whose centre is the origin, of radius ρ, there are no points ζ representing values corresponding to |z| < R. But if ζ₀ be the value of ζ corresponding to (x₀, y₀), and the expression of ζ − ζ₀ 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 π/m + i sin π/m), will correspond two points near to ζ₀, say ζ₁, and 2ζ₀ − ζ′₁, situated so that ζ₀ is between them. One of these must be within the circle (ρ). We infer then that ρ＝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 − α, where α denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

§ 3. Limiting Operations.—In order that a complex number ζ＝ξ + iη may have a limit it is necessary and sufficient that each of ξ and η 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 ε, 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 ε 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 ε 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_nzⁿ, 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_nz₀ⁿ| < 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 Σa_nzⁿ, Σb_nzⁿ 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 Σa_n 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, |a_n| < Mr₁⁻ⁿ. If in every arbitrarily small neighbourhood of z=0 there be a point for which two converging power series Σa_n zⁿ, Σb_nzⁿ agree in value, then the series are identical, or a_n＝b_n; thus also if Σa_nzⁿ 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)＝Σa_nzⁿ of radius of convergence R, if |z₀| < R and we put z＝z₀ + t with |t| < R-|z₀|, the resulting series Σa_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 ƒ(z)＝Σ A_ntⁿ; hence A₀＝ƒ(z₀), and we have [ƒ(z₀ + t) − ƒ(z₀)]/t = Σn=1 A_nt 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 Σna_nz₀ⁿ⁻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)＝Σtⁿƒ⁽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 Σa_nzⁿ, 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 e^z＝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＝1/2y² − 1 + cos y, U₂＝1/6y ³ − y + sin y, V₂＝1/24y 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 + 1/24y ⁴ − . . .,   sin y＝y − 1/6y ³ + 1/120y 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 + 2πi)＝exp (x) [cos (y+2π) + i sin(y+2π)],

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

The function λ(ζ) is required to define ζ^a when ζ and a are complex numbers; this is defined as exp [aλ(ζ)], that is as Σn=0 aⁿ [λ(ζ)]ⁿ/n!. When a is a real integer the ambiguity of λ(ζ) is immaterial here, since exp [aλ(ζ)+2kaπi]＝exp [aλ(ζ)]; when a is of the form 1/q, where q is a positive integer, there are q values possible for ζ^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 ζ; when a＝p/q, where p, q are integers without common factor, q being positive, we have ζ^p/q＝(ζ^1/q)^p. The definition of the symbol ζ^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 1/2π; thus λ(i)＝i (1/2π+2kπ); thus

i ⁱ＝exp (−1/2π − 2kπ)＝exp (−1/2π) exp (−2kπ),

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