225. The general power aζ.

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

We have already defined the meaning of the symbol aζ in a considerable variety of cases. It is defined in elementary Algebra in the case in which a is real and positive and ζ rational, or a real and negative and ζ a rational fraction whose denominator is odd. According to the definitions there given aζ 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 ζ any rational number p/q; and in Ch. IX we gave a new definition, expressed by the equation aζ=eζloga, which applies whenever ζ is real and a real and positive.

Thus we have, in one way or another, attached a meaning to such expressions as 31/2,(1)1/3,(3+12i)1/2,(3.5)1+2; but we have as yet given no definitions which enable us to attach any meaning to such expressions as (1+i)2,2i,(3+2i)2+3i. We shall now give a general definition of aζ which applies to all values of a and ζ, real or complex, with the one limitation that a must not be equal to zero.

Definition. The function aζ is defined by the equation aζ=exp(ζloga) where loga 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 ζ real, then one value of ζloga, viz. ζloga, is real: and exp(ζloga)=eζloga, 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=eτ(cosψ+isinψ), then loga=τ+i(ψ+2mπ),exp{(p/q)loga}=epτ/qCis{(p/q)(ψ+2mπ)}, 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 ap/q found in § 48. Hence our new definition is also consistent with that of Ch. III.

 

226. The general value of aζ.

Let ζ=ξ+iη,a=σ(cosψ+isinψ) where π<ψπ, so that, in the notation of § 225, σ=eτ or τ=logσ.

Then ζloga=(ξ+iη){logσ+i(ψ+2mπ)}=L+iM, where L=ξlogση(ψ+2mπ),M=ηlogσ+ξ(ψ+2mπ); and aζ=exp(ζloga)=eL(cosM+isinM). Thus the general value of aζ is eξlogση(ψ+2mπ)[cos{ηlogσ+ξ(ψ+2mπ)}+isin{ηlogσ+ξ(ψ+2mπ)}].

In general aζ is an infinitely many-valued function. For |aζ|=eξlogση(ψ+2mπ) has a different value for every value of m, unless η=0. If on the other hand η=0, then the moduli of all the different values of aζ are the same. But any two values differ unless their amplitudes are the same or differ by a multiple of 2π. This requires that ξ(ψ+2mπ) and ξ(ψ+2nπ), where m and n are different integers, shall differ, if at all, by a multiple of 2π. But if ξ(ψ+2mπ)ξ(ψ+2nπ)=2kπ, then ξ=k/(mn) is rational. We conclude that aζ is infinitely many-valued unless ζ is real and rational. On the other hand we have already seen that, when ζ is real and rational, aζ has but a finite number of values.

The principal value of aζ=exp(ζloga) is obtained by giving loga its principal value, by supposing m=0 in the general formula. Thus the principal value of aζ is eξlogσηψ{cos(ηlogσ+ξψ)+isin(ηlogσ+ξψ)}.

Two particular cases are of especial interest. If a is real and positive and ζ real, then σ=a, ψ=0, ξ=ζ, η=0, and the principal value of aζ is eζloga, which is the value defined in the last chapter. If |a|=1 and ζ is real, then σ=1, ξ=ζ, η=0, and the principal value of (cosψ+isinψ)ζ is cosζψ+isinζψ. This is a further generalisation of De Moivre’s Theorem (§§ 45, 49).

Example XCIV
1. Find all the values of ii. [By definition ii=exp(ilogi). But i=cos12π+isin12π,logi=(2k+12)πi, where k is any integer. Hence ii=exp{(2k+12)π}=e(2k+12)π. All the values of ii are therefore real and positive.]

2. Find all the values of (1+i)i, i1+i, (1+i)1+i.

3. The values of aζ, 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 aζ=r(cosθ+isinθ) we have r=eξlogση(ψ+2mπ),θ=ηlogσ+ξ(ψ+2mπ); and all the points lie on the spiral r=σ(ξ2+η2)/ξeηθ/ξ.]

4. The function eζ. If we write e for a in the general formula, so that logσ=1, ψ=0, we obtain eζ=eξ2mπη{cos(η+2mπξ)+isin(η+2mπξ)}. The principal value of eζ is eξ(cosη+isinη), which is equal to expζ (§ 223). In particular, if ζ is real, so that η=0, we obtain eζ(cos2mπζ+isin2mπζ) as the general and eζ as the principal value, eζ denoting here the positive value of the exponential defined in Ch. IX.

5. Show that logeζ=(1+2mπi)ζ+2nπi, where m and n are any integers, and that in general logaζ has a double infinity of values.

6. The equation 1/aζ=aζ is completely true (Ex. XCIII. 3): it is also true of the principal values.

7. The equation aζ×bζ=(ab)ζ is completely true but not always true of the principal values.

8. The equation aζ×aζ=aζ+ζ 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 aζ×aζ, viz. exp{ζ(loga+2mπi)+ζ(loga+2nπi)}, is not as a rule a value of aζ+ζ unless m=n.]

9. What are the corresponding results as regards the equations logaζ=ζloga,(aζ)ζ=(aζ)ζ=aζζ?

10. For what values of ζ is (a) any value (b) the principal value of eζ (i) real (ii) purely imaginary (iii) of unit modulus?

11. The necessary and sufficient conditions that all the values of aζ should be real are that 2ξ and {ηlog|a|+ξama}/π, where ama 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 |xi+xi|, where x>0, is e(mn)π2{cosh2(m+n)π+cos(2logx)}.

13. Explain the fallacy in the following argument: since e2mπi=e2nπi=1, where m and n are any integers, therefore, raising each side to the power i we obtain e2mπ=e2nπ.

14. In what circumstances are any of the values of xx, where x is real, themselves real? [If x>0 then xx=exp(xlogx)=exp(xlogx)Cis2mπx, 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(xlogx), given by m=q.

If x=ξ<0 then xx=exp{ξlog(ξ)}=exp(ξlogξ)Cis{(2m+1)πξ}. The only case in which any value is real is that in which ξ=p/(2q+1), when m=q gives the real value exp(ξlogξ)Cis(pπ)=(1)pξξ. The cases of reality are illustrated by the examples (13)1/3=133,(12)12=±12,(23)23=943,(13)13=33.]

15. Logarithms to any base. We may define ζ=logaz in two different ways. We may say (i) that ζ=logaz if the principal value of aζ is equal to z; or we may say (ii) that ζ=logaz if any value of aζ is equal to z.

Thus if a=e then ζ=logez, according to the first definition, if the principal value of eζ is equal to z, or if expζ=z; and so logez is identical with logz. But, according to the second definition, ζ=logez if eζ=exp(ζloge)=z,ζloge=logz, or ζ=(logz)/(loge), any values of the logarithms being taken. Thus ζ=logez=log|z|+(amz+2mπ)i1+2nπi, so that ζ is a doubly infinitely many-valued function of z. And generally, according to this definition, logaz=(logz)/(loga).

16. loge1=2mπi/(1+2nπi), loge(1)=(2m+1)πi/(1+2nπi), where m and n are any integers.


222–224. The exponential function Main Page 227–230. The trigonometrical and hyperbolic functions