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(;<±r) + /(»±f) = (a + iv) ± (r -f ?'.?)
= w ±

Thus the law of indices obtains for complex quantities.
(Cf. Art. 27.)

79. The Addition Theorem. Operating upon both sides of the equation last written with logK, we have

logK {BW Y/ B>) = w ± t,

which, if Bw = z and Bl = z', and therefore z logKw and 2'= log*/, becomes

logK O >^ ) = logK z ± log*

which is the addition theorem for complex quantities. (Cf. Art. 28.)

80. The Logarithmic Spiral. The locus of P, in Fig. 36 of Art. 68, is obviously a spiral. Its polar equation may be obtained by considering the rates of change of p and 6, which are respectively \p and a> = Xtan(<£— /3). By the definition of a logarithm for real quantities (Art. 23), 9 is here the logarithm of p with respect to the modulus X tan (<t> — /3) / X = tan (<£ — ,3 ); or in equivalent terms,

0 = tan O — /3) -Inp,

which is the equation sought.

This locus is called the logarithmic spiral. It is obvious from the definition of the motion of the point that generates this curve (Art. 68), that the spiral encircles the origin an infinite number of times, coming nearer to it with every return, and that it likewise passes around the circumference of the unit circle an infinite number of times, going continually farther away from it.

81. Periodicity of Exponentials. From its character as an operator that turns any line in the plane through the angle whose arc-ratio is 6, it is evident that cis 6 acquires once all its possible values corresponding to real values of 6, when 6 passes continuously from o to 2ir and, k being any integer, repeats the same cycle of values through every interval from 2 kit to 2(£ + i)ir, so that for all integral values of k,

cis (6 + 2 kit) = cis 6.

In consequence of this property, cis0 is said to be periodic, having the period 2 jr.

The exponential BTM has similarly a period. Solving for ?t and v the two linear eqations,

«'= u cos B + v sin /3,

v'— v cos j3—7«sin/3, (Art. 73.)

we easily find

u = u' cos & v' sin $
v = u' sin P + 1/ cos jS;

whence

-j- iv = («'+ zV) cis ,3;

or this result may be inferred by direct inspection of Fig. 36. Hence the equation £u+'v = bu' cisv'/m, of Art. 73, may be written ,

^("^"Osiscis

m

In this formula let «'= b and v'= 2kmir, k being any integer. Then, since cis 2kir= 1,

wz

and therefore

Hence Bw has the period 2zktt. In particular ew has the period 2ztt.

It is obvious that the only series of values of v' that will render cisv'/m=i is zkmir, where k is an integer (Art. 63). Hence the only value of w, that will render gw+w, fiw ;s W' = 2kiKir. 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

logK z = W -f- 2lkKir,

that is, \ogKz, for a given value of z, has an indefinitely great number of values, differing from each other, in successive pairs, by 2zW. 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,

ln^= w -f- 2ik.Tr.

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 f (or some other letter) stand for any one of the direct functional symbols used, such as £, or b, thus expressing z as a direct function of w in the form

z=f(w),

and then writing

for the purpose of expressing the fact that w is the corresponding inverse function of z. Thus when Bw takes the place ofy(w), log^ z takes the place of f~' (0).

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 /-% thus:

f-\z)=f->f(w) = w.

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

(i) . Addition and subtraction:

(x -f- a) a = x, (x a) -\- a = x.

(ii) . Multiplication and division:

(jc X *) / « = x, (x / a) X « = *,

(iii) . Exponentiation and logarithmic operation:

"log a* = x, a x = x.

(iv) . Involution and evolution:

(xayl« = x, (xlfa)a=x.

In the use of the notation of inversion here described, its application to symbolic operation, in the form ff~l (w) = i0, must be carefully distinguished from its application to products and quotients by which from ab = c is derived a = b~xc.

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

f(w + nj>) =/<» = 2,

then

/- (2) = w + np, for all integral values ofw. (Cf. Art. 82.)

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