225–226. The general power $a^z$

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).

Example XCIV
1. Find all the values of \(i^{i}\). [By definition \[i^{i} = \exp (i\log i).\] But \[i = \cos \tfrac{1}{2}\pi + i\sin \tfrac{1}{2}\pi,\quad \log i = (2k + \tfrac{1}{2})\pi i,\] where \(k\) is any integer. Hence \[i^{i} = \exp\{-(2k + \tfrac{1}{2})\pi\} = e^{-(2k + \frac{1}{2})\pi}.\] All the values of \(i^{i}\) are therefore real and positive.]

2. Find all the values of \((1 + i)^{i}\), \(i^{1+i}\), \((1 + i)^{1+i}\).

3. The values of \(a^{\zeta}\), when plotted in the Argand diagram, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of \(a\).

[If \(a^{\zeta} = r(\cos\theta + i\sin\theta)\) we have \[r = e^{\xi\log \sigma – \eta(\psi + 2m\pi)},\quad \theta = \eta\log \sigma + \xi(\psi + 2m\pi);\] and all the points lie on the spiral \(r = \sigma^{(\xi^{2} + \eta^{2})/\xi} e^{-\eta \theta/\xi}\).]

4. The function \(e^{\zeta}\). If we write \(e\) for \(a\) in the general formula, so that \(\log \sigma = 1\), \(\psi = 0\), we obtain \[e^{\zeta} = e^{\xi-2m\pi\eta} \{\cos(\eta + 2m\pi\xi) + i\sin(\eta + 2m\pi\xi)\}.\] The principal value of \(e^{\zeta}\) is \(e^{\xi}(\cos\eta + i\sin\eta)\), which is equal to \(\exp \zeta\) (§ 223). In particular, if \(\zeta\) is real, so that \(\eta = 0\), we obtain \[e^{\zeta} (\cos 2m\pi\zeta + i\sin 2m\pi\zeta)\] as the general and \(e^{\zeta}\) as the principal value, \(e^{\zeta}\) denoting here the positive value of the exponential defined in Ch. IX.

5. Show that \(\log e^{\zeta} = (1 + 2m\pi i)\zeta + 2n\pi i\), where \(m\) and \(n\) are any integers, and that in general \(\log a^{\zeta}\) has a double infinity of values.

6. The equation \(1/a^{\zeta} = a^{-\zeta}\) is completely true (Ex. XCIII. 3): it is also true of the principal values.

7. The equation \(a^{\zeta} \times b^{\zeta} = (ab)^{\zeta}\) is completely true but not always true of the principal values.

8. The equation \(a^{\zeta} \times a^{\zeta’} = a^{\zeta+\zeta’}\) is not completely true, but is true of the principal values. [Every value of the right-hand side is a value of the left-hand side, but the general value of \(a^{\zeta} \times a^{\zeta’}\), viz. \[\exp \{\zeta(\log a + 2m\pi i) + \zeta'(\log a + 2n\pi i)\},\] is not as a rule a value of \(a^{\zeta+\zeta’}\) unless \(m = n\).]

9. What are the corresponding results as regards the equations \[\log a^{\zeta} = \zeta\log a,\quad (a^{\zeta})^{\zeta’} = (a^{\zeta’})^{\zeta} = a^{\zeta\zeta’}?\]

10. For what values of \(\zeta\) is (a) any value (b) the principal value of \(e^{\zeta}\) (i) real (ii) purely imaginary (iii) of unit modulus?

11. The necessary and sufficient conditions that all the values of \(a^{\zeta}\) should be real are that \(2\xi\) and \(\{\eta\log |a| + \xi\operatorname{am} a\}/\pi\), where \(\operatorname{am} a\) denotes any value of the amplitude, should both be integral. What are the corresponding conditions that all the values should be of unit modulus?

12. The general value of \(|x^{i} + x^{-i}|\), where \(x > 0\), is \[e^{-(m-n)\pi} \sqrt{2\{\cosh 2(m + n)\pi + \cos(2\log x)\}}.\]

13. Explain the fallacy in the following argument: since \(e^{2m\pi i} = e^{2n\pi i} = 1\), where \(m\) and \(n\) are any integers, therefore, raising each side to the power \(i\) we obtain \(e^{-2m\pi} = e^{-2n\pi}\).

14. In what circumstances are any of the values of \(x^{x}\), where \(x\) is real, themselves real? [If \(x > 0\) then \[x^{x} = \exp (x\log x) = \exp (x\log x) \operatorname{Cis} 2m\pi x,\] the first factor being real. The principal value, for which \(m = 0\), is always real.

If \(x\) is a rational fraction \(p/(2q + 1)\), or is irrational, then there is no other real value. But if \(x\) is of the form \(p/2q\), then there is one other real value, viz. \(-\exp (x\log x)\), given by \(m = q\).

If \(x = -\xi < 0\) then \[x^{x} = \exp \{-\xi\log (-\xi)\} = \exp (-\xi\log \xi) \operatorname{Cis}\{-(2m + 1)\pi\xi\}.\] The only case in which any value is real is that in which \(\xi = p/(2q + 1)\), when \(m = q\) gives the real value \[\exp (-\xi\log \xi) \operatorname{Cis} (-p\pi) = (-1)^{p} \xi^{-\xi}.\] The cases of reality are illustrated by the examples \[(\tfrac{1}{3})^{1/3} = \sqrt[3]{\tfrac{1}{3}},\quad (\tfrac{1}{2})^{\frac{1}{2}} = \pm\sqrt{\tfrac{1}{2}},\quad (-\tfrac{2}{3})^{-\frac{2}{3}} = \sqrt[3]{\tfrac{9}{4}},\quad (-\tfrac{1}{3})^{-\frac{1}{3}} = -\sqrt[3]{3}.]\]

15. Logarithms to any base. We may define \(\zeta = \log_{a} z\) in two different ways. We may say (i) that \(\zeta = \log_{a} z\) if the principal value of \(a^{\zeta}\) is equal to \(z\); or we may say (ii) that \(\zeta = \log_{a} z\) if any value of \(a^{\zeta}\) is equal to \(z\).

Thus if \(a = e\) then \(\zeta = \log_{e} z\), according to the first definition, if the principal value of \(e^{\zeta}\) is equal to \(z\), or if \(\exp \zeta = z\); and so \(\log_{e} z\) is identical with \(\log z\). But, according to the second definition, \(\zeta = \log_{e} z\) if \[e^{\zeta} = \exp (\zeta\log e) = z,\quad \zeta\log e = \log z,\] or \(\zeta = (\log z)/(\log e)\), any values of the logarithms being taken. Thus \[\zeta = \log_{e} z = \frac{\log |z| + (\operatorname{am} z + 2m\pi)i}{1 + 2n\pi i},\] so that \(\zeta\) is a doubly infinitely many-valued function of \(z\). And generally, according to this definition, \(\log_{a} z = (\log z)/(\log a)\).

16. \(\log_{e} 1 = 2m\pi i/(1 + 2n\pi i)\), \(\log_{e}(-1) = (2m + 1)\pi i/(1 + 2n\pi i)\), where \(m\) and \(n\) are any integers.

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