m=

It is obvious that the only series of values of ʊ that will render cis v'/m 1 is 2km, where k is an integer (Art. 63). Hence the only value of w, that will render Bw+w, Bw is w,= 2kiкπ. The function Bw therefore has only one period and is said to be singly periodic.

=

82. Many-Valuedness of Logarithms. As a consequence of the periodicity of Bw=z, the logarithm (for all integral values of k) has the form

logw2ikкπ,

that is, log, for a given value of z, has an indefinitely great number of values, differing from each other, in successive pairs, by 2iкя. The logarithm is therefore said to be many-valued; specifically its many-valuedness is infinite. We shall discover this property in other functions. In the natural agonic system,

83. Direct and Inverse Processes. In the present section and in Sec. VII we have had occasion to speak of logarithms and exponentials as inverse to one another. More explicitly, the exponential is called the direct function, the logarithm its inverse. We express the operation of inversion in general terms by letting ƒ (or some other letter) stand for any one of the direct functional symbols used, such as B, or b, thus expressing z as a direct function of w in the form

and then writing

z=f(w),

for the purpose of expressing the fact that w is the corresponding inverse function of z. Thus when Bw takes the place of f(w), log takes the place of ƒ ̃1(2).

In other terms, inversion is described as that process which annuls the effect of the direct process. If z=f(w), the effect of the operation f is annulled by the operation f, thus:

f(z) =ƒ'ƒ(w)=w.

In accordance with this definition the following processes are inverse to each other:

(i). Addition and subtraction:

(x + a) ·

a= = x, (x − a)+a=x.

(ii). Multiplication and division:

(xa) a = x, (x/a) Xa === x,

In the use of the notation of inversion here described, its application to symbolic operation, in the form fƒ—1 (w)=w, must be carefully distinguished from its application to products and quotients by which from abc is derived a=b-c.

Whenever the direct function is periodic its inverse is obviously many-valued; for, if p be a period of ƒ (w), so that

then

f(w+np)=f(w)=z,

ƒ1 (z) = w+np,

for all integral values of n.

(Cf. Art. 82.)

84. Agenda. Reduction of Exponential and Logarithmic Forms. Prove the following:

(1).

(a+b)u+iv

= cu / 2 • In (a2+b2)—v tan ̄1 b/a cis [vln (a2+b2) + u tan¬1b/a].

(2). (ib)u+iveunb-v/2 cis (vlnbu).

(3). ¿u+iv υπ/ 2 cis ήπι.

ii =

=

(4). e-/2, i2i/= c.

+

(5). logm+in (x + iy) = 1 m ln (x2 + y2) — n tan ̄1y/x

+i[n]n (x2 + y2)+m tan-1y/x].

(6). logm+in iy=mlny - n+i (n lny + 1⁄2 mπ).

(7). log(x)= in x-.

(8). log(x)=- Inx - in.

(9). log;i=— 1π, log; (— i) = { π.

(10). Given a + ib as the base of a system of logarithms, find the modulus and reduce it to the form u + iv.

(11). Express a+iblog (x + y) in the form u + iv in terms of a, b, x and y.

(12). If u'=u cos ß + v sin ß, v = cos ẞu sin ß, w=uiv and κm (cos ẞ+ i sin ẞ), prove:

(13). From f(w)= aw2 + 2bw+c=z deduce

ƒ1(2)=(b±y b2 — ac + az)/a.

XIII. WHAT CONSTITUTES AN ALGEBRA?

85. The Cycle of Operations Complete. It is obvious from the manner in which in the preceding pages the algebraic processes as applied to complex quantities have been defined, that out of their operation in any possible combination no new forms of quantity can arise. But in particular the examples of Art. 67 illustrate this fact in an explicit way in the case of geometric addition and multiplication; and the exponential equation

or

K

(Art. 73.)

(Art. 75.)

x + iy

√x2 + y2

(tan-2),

; ln (x2 + y2) + ln cis (tan

log× (x + iy) = 1⁄2 În (x2 + y2) + i « tan−1?

2

y

x

exhibit with equal clearness that only complex quantities can result from the application of exponential and logarithmic operations. Hence the processes of addition and subtraction, multiplication and division, involution and evolution, exponentiation and the taking of logarithms complete the cycle of operations necessary to algebra. Other operations indeed will be introduced and applied to complex quantities, such as those defined in Art. 90, but though expressed in abbreviated form as single operations, they are combinations of those already described.

86. Definition of an Algebra. When a series of elements operating upon each other in accordance with fixed laws produce only other elements belonging to the same series, they are said to constitute a group. Thus all positive integers, subject only to the processes of addition and multiplication, produce only positive integers, and hence form a group. Such a group is an algebra.

The effect of introducing into the arithmetic of positive integers the further processes of subtraction and division is to break the integrity of the old group and form a new one whose elements include, not only positive integers, but all rational numbers, both positive and negative, integral and fractional. A final step through evolution, or the extracting of roots, bringing with it the logarithmic operation, leads to imaginary and complex numbers—that is, numbers composed of both a real and an imaginary part.

If now, as is legitimate, we regard all reals and imaginaries as special forms of complex quantities-reals having zero imaginary parts, imaginaries having zero real partsthen, as pointed out in the preceding article, the algebraic processes of addition, subtraction, multiplication, division, evolution, involution, exponentiation, and the taking of logarithms (logarithmication), applied to complex quantities in any of their several forms, produce only other complex quantities. And hence :

The aggregate of all complex quantities — including all reals and imaginaries, both rational and irrational — operating upon each other in all possible ways by the rules of algebra, form a closed group. Such a group is again an algebra.

If, in an algebra the elements that constitute the subjects of its operations form a closed group when subjected to a complete cycle of such operations, such an algebra