## 225. The general power \(a^{\zeta}\).

It might seem natural, as \(\exp \zeta = e^{\zeta}\) when \(\zeta\) is real, to adopt the same notation when \(\zeta\) is complex and to drop the notation \(\exp \zeta\) altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol \(e^{\zeta}\): we shall find then that \(e^{\zeta}\) represents a function with infinitely many values of which \(\exp \zeta\) is only one.

We have already defined the meaning of the symbol \(a^{\zeta}\) in a considerable variety of cases. It is defined in elementary Algebra in the case in which \(a\) is real and positive and \(\zeta\) rational, or \(a\) real and negative and \(\zeta\) a rational fraction whose denominator is odd. According to the definitions there given \(a^{\zeta}\) has at most two values. In Ch. III we extended our definitions to cover the case in which \(a\) is any real or complex number and \(\zeta\) any rational number \(p/q\); and in Ch. IX we gave a new definition, expressed by the equation \[a^{\zeta} = e^{\zeta\log a},\] which applies whenever \(\zeta\) is real and \(a\) real and positive.

Thus we have, in one way or another, attached a meaning to such expressions as \[3^{1/2},\quad (-1)^{1/3},\quad (\sqrt{3} + \tfrac{1}{2}i)^{-1/2},\quad (3.5)^{1+\sqrt{2}};\] but we have as yet given no definitions which enable us to attach any meaning to such expressions as \[(1 + i)^{\sqrt{2}},\quad 2^{i},\quad (3 + 2i)^{2+3i}.\] We shall now give a general definition of \(a^{\zeta}\) which applies to all values of \(a\) and \(\zeta\), real or complex, with the one limitation that \(a\) must not be equal to zero.

**Definition.**

*The function \(a^{\zeta}\) is defined by the equation \[a^{\zeta} = \exp (\zeta\log a)\] where \(\log a\) is any value of the logarithm of \(a\).*

We must first satisfy ourselves that this definition is consistent with the previous definitions and includes them all as particular cases.

(1) If \(a\) is positive and \(\zeta\) real, then one value of \(\zeta\log a\), viz. \(\zeta\log a\), is real: and \(\exp (\zeta\log a) = e^{\zeta\log a}\), which agrees with the definition adopted in Ch. IX. The definition of Ch. IX is, as we saw then, consistent with the definition given in elementary Algebra; and so our new definition is so too.

(2) If \(a = e^{\tau} (\cos\psi + i\sin\psi)\), then \[\begin{gathered} \log a = \tau + i(\psi + 2m\pi), \\ \exp \{(p/q)\log a\} = e^{p\tau/q} \operatorname{Cis} \{(p/q)(\psi + 2m\pi)\},\end{gathered}\] where \(m\) may have any integral value. It is easy to see that if \(m\) assumes all possible integral values then this expression assumes \(q\) and only \(q\) different values, which are precisely the values of \(a^{p/q}\) found in § 48. Hence our new definition is also consistent with that of Ch. III.

## 226. The general value of \(a^\zeta\).

Let \[\zeta = \xi + i\eta,\quad a = \sigma(\cos\psi + i\sin\psi)\] where \(-\pi < \psi \leq \pi\), so that, in the notation of § 225, \(\sigma = e^{\tau}\) or \(\tau = \log \sigma\).

Then \[\zeta \log a = (\xi + i\eta)\{\log \sigma + i(\psi + 2m\pi)\} = L + iM,\] where \[L = \xi \log \sigma – \eta(\psi + 2m\pi),\quad M = \eta\log \sigma + \xi (\psi + 2m\pi);\] and \[a^{\zeta} = \exp(\zeta\log a) = e^{L}(\cos M + i\sin M).\] Thus the general value of \(a^{\zeta}\) is \[e^{\xi\log \sigma – \eta(\psi+2m\pi)} [\cos\{\eta\log \sigma + \xi(\psi + 2m\pi)\} + i\sin\{\eta\log \sigma + \xi(\psi + 2m\pi)\}].\]

In general \(a^{\zeta}\) is an infinitely many-valued function. For \[|a^{\zeta}| = e^{\xi\log \sigma – \eta(\psi+2m\pi)}\] has a different value for every value of \(m\), unless \(\eta = 0\). If on the other hand \(\eta = 0\), then the moduli of all the different values of \(a^{\zeta}\) are the same. But any two values differ unless their amplitudes are the same or differ by a multiple of \(2\pi\). This requires that \(\xi(\psi + 2m\pi)\) and \(\xi(\psi + 2n\pi)\), where \(m\) and \(n\) are different integers, shall differ, if at all, by a multiple of \(2\pi\). But if \[\xi(\psi + 2m\pi) – \xi(\psi + 2n\pi) = 2k\pi,\] then \(\xi = k/(m – n)\) is rational. We conclude that *\(a^{\zeta}\) is infinitely many-valued unless \(\zeta\) is real and rational*. On the other hand we have already seen that, when \(\zeta\) is real and rational, \(a^{\zeta}\) has but a finite number of values.

The *principal value* of \(a^{\zeta} = \exp (\zeta\log a)\) is obtained by giving \(\log a\) its principal value, by supposing \(m = 0\) in the general formula. Thus the principal value of \(a^{\zeta}\) is \[e^{\xi\log \sigma – \eta\psi} \{\cos(\eta\log \sigma + \xi\psi) + i\sin(\eta\log \sigma + \xi\psi)\}.\]

Two particular cases are of especial interest. If \(a\) is real and positive and \(\zeta\) real, then \(\sigma = a\), \(\psi = 0\), \(\xi = \zeta\), \(\eta = 0\), and the principal value of \(a^{\zeta}\) is \(e^{\zeta\log a}\), which is the value defined in the last chapter. If \(|a| = 1\) and \(\zeta\) is real, then \(\sigma = 1\), \(\xi = \zeta\), \(\eta = 0\), and the principal value of \((\cos\psi + i\sin\psi)^{\zeta}\) is \(\cos\zeta\psi + i\sin\zeta\psi\). This is a further generalisation of De Moivre’s Theorem (§§ 45, 49).

$\leftarrow$ 222–224. The exponential function | Main Page | 227–230. The trigonometrical and hyperbolic functions $\rightarrow$ |