cos y = 1 − 1⁄2y² + 1⁄24y4 − ..., sin y = y − 1⁄6y³ + 1⁄120y5 − ...,
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 − ½(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 ζa 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 ζ1/q, of the form exp [1/q λ(ζ)] exp (2kπi/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 ζ; when a = p/q, where p, q are integers without common factor, q being positive, we have ζp/q = (ζ1/q)p. The definition of the symbol ζa 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 ii; the number i is of modulus unity and phase ½π; thus λ(i) = i (½π + 2kπ); thus
ii = exp (−½π − 2kπ) = exp (−½π) exp (−2kπ),
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 = ½[exp (iz) + exp (−iz)] and sin z = −½i [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.