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.