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

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227–230. The trigonometrical and hyperbolic functions $\rightarrow$

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

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