In these expressions Ab means any solution of the equation obtained by equating the expansional value of e* to b: i.e. λb represents some logarithm of b to the base e. X 648. The indeterminateness of these operations may be diminished however by the following conventions : (1) The written symbol of the form e shall mean the expansional value of e. (2) The written symbol of the form log, a shall mean any value of x for which the expansional value of e* = ɑ. and Then from the equation a = b, we should deduce a2 = e(logeb+2nπî)x log,a= log b + 2nπi. In this way the logarithm to the base e would be reduced to a single indeterminateness. But those to any other base would have to remain doubly indeterminate. х 649. Illustrations may be given of the confusions arising from neglecting the nature of the indeterminateness of powers and logarithms. 1o = ¿¿ logo1 – the expansional value of e3(0+2nπi) =e = = the expansional value of e-2. This is an indeterminate result one of whose values is 1, Next we have a value of e2i is 1; .. a value of e viz. i.e. the expansional value of e for some value of n; expansional value of e Raise both sides to the power 1 + 2nπi, then e1−4mnπ2+2(m+n)wi 1+2nπi = for any integral values of m and n. =e = e, This process might be repeated ad infinitum. It must be amended as follows: For simplicity we pansional value of e. assigned value of m. may understand that e" means the ex 1+2mπi Then we may write e' = e, for any Raising both sides of this equation to the power 1 + 2nπi and still understanding expansional values, the left hand becomes (1+2m#i+2m'wi) (1+2ni) where m' is unassigned. e Assigning to m' its several values we obtain the several values required. Amongst these, viz. when m'-m, we have the value e. Applications to Trigonometrical Formulae. 650. Having worked out a purely algebraical theory of imaginary indices or logarithms we may, by means of De Moivre's Theorem, apply the results to Trigonometry. 651. To find a series for cos A + i sin A. By De Moivre's Theorem, if A is a real angle and x a real quantity, the general values of (cos A+ i sin A) are given by the expression cos x ( A + n. 360°) + i sin x (A + n. 360°), where n is any integer. But, by the theorems above discussed, the general values of (cos A+ i sin A) are the values of Hence, if c satisfies (1), cos x (A + n. 360°) + i sin x (A + n. 360°) = 1 + cx + Now x, being fractional, may be indefinitely diminished. and sin x (A + n . 360°) – circular measure of A + n. 360° ; = .. from (2), i × (circular measure of A + n. 360°) = c. Let 0 and 2π be the circular measure of A and 360° respectively. Then by (1), 641. The expansional value of ei (0 +2nπ) = cos 0+ i sin .(3). This result justifies the introduction of the symbol π in Art. 652. If we understand by e its expansional value, we have cos + i sin 0 = e and .. cos - i sin 0 = e ̄io .. cos 0 = § (e1o + e ̄1o) and i sin 0 = 1 (e1o − e−io). These are usually called the Exponential values of the sine and cosine. It would be more correct to call cos and i sin the Ex io pansional values of (e + e ̄) and (e1o — e-io) respectively. 1⁄2 For these latter are at first sight uninterpretable, and are in any case indeterminate; while cos◊ and sin are both interpretable and determinate. 653. We have already (Art. 508) given the following definitions: cosh 0 = the expansional value of (eo + e ̃o), and Hence we may introduce imaginary angles, by the definitions: cos i cosh 0 and sin i = i sinh 0; = which will result from our writing i0 for 0 in the values of cos and i sin 0. Thus a wholly imaginary angle has a wholly real cosine, but a wholly imaginary sine. But since here e means the expansional value of e, we may use log, a to mean a value of x for which the expansional value of ex is a. Hence, if tan <1, 0 But the series after na is equal to the value of ◊ between – 1π and by Art. 645. Or, immediately by the principle of continuity, since tan 0=0; we have = the circular measure of the acute value of tan-1 x. 655. The exponential expressions for the sine and cosine enable us to use very compendious proofs of propositions (which, of course, are otherwise provable). They have two important uses : (1) In the interpretation of functions involving complex expressions: i.e. the reduction of these functions to the form of a complex expression. (2) In the summation of series and conversely the development of functions into series. 656. Example of the interpretation of a complex function. Reduce Let a + bi Then (a + bi)p+qi to the form A + Bi. Ө = r (cos 0 + i sin 0), so that r2 = a2 + b2 and tan 0 = b/a. (a+bi)p+qi = pp+qi ̧e(p+qi)oi = pP ̧e ̄9o ̧rgi ¿poi .. A = r2. e ̄90 = r2.e-go.e(a log r+pe) i go = pp. cos (q logr + p0) and B = r2 e sin (q log r + p0). The indeterminateness of this result should be examined. We take r to be the positive value of √(a2 + b2): so that to logr we may give its real value. Then is the angle whose cosine is b/r and whose sine is a/r. Hence is indeterminate-its values differing by 2π. Hence the magnitude of the coefficient eq' has an infinite number of values; and the expression cos (q log r+p8) + i sin (q log r+pė) has values equal in number to the denominator of p. 657. Example of the Summation of Series. To find the two sums: We have C=1+x cos 0 + x2 cos 20 + x3 cos 30 + ... C + Si = 1 + x (cos 0 + i sin 0) + x2 (cos 20 + i sin 20) + 658. Example of Development in a Series. To expand log, (1 - 2x cos + x2) in powers of x. Log. (1-2x cos 0+x2)=log. (1-xei-xe-i+x2)=log. (1-xe°)(1–xe ̄i) 0+x2)=log,(1−xe" == 20i x (eoi + e ̄0i) — § x2 (e2oi + e−20i) — —ï3 (e3 30i +e -30i) - 1⁄2 log. (1 − 2x cos 0 + x2) = x cos 0 + 1x2 cos 20 + 1x3 cos 30 + ... ) |