In these expressions Ab means any solution of the equation obtained by equating the expansional value of e to b: i.e. b 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 ea.

(2) The written symbol of the form log, a shall mean any value of x for which the expansional value of e*

and

Then from the equation a=b, we should deduce

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.

i

1=log1 = the expansional value of e1(0+2ní)

This is an indeterminate result one of whose values is 1, viz.

when n =
= 0.

Next we have a value of ei is 1; .. a value of e

for some value of n; i.e. the expansional value of e expansional value of e2n" if n = 1.

Raise both sides to the power 1 + 2nπi, then

e1−4mnπ2+2(m+n) wi = e1+2nwi = e,

for any integral values of m and n.

This process might be repeated ad infinitum.

It must be amended as follows:

For simplicity we may understand that e" means the expansional value of e*.

assigned value of m.

1+2mπi

Then we may write e

=

= l,

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'πi) (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 (4 + n. 360°) + i sin x (A + n. 360°) = 1 + cx +

Now x, being fractional, may be indefinitely diminished.

.. from (2), i × (circular measure of A +n. 360°) = c.

Let 0 and 2 be the circular measure of A and 360° respectively. Then by (1),

641.

The expansional value of

e(0+2
+2nπ)

=

cos + 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

i0

cos + i sin 0 =¿ and .. cos - i sin 0 = e-io.

=e

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

е

pansional values of (e + e ̄io) and 1⁄2 (e1o − e−1o) respectively. For these latter are at first sight uninterpretable, and are in any case indeterminate; while cos 0 and sin are both interpretable and determinate.

653. We have already (Art. 508) given the following definitions :

Hence we may introduce imaginary angles, by the definitions:

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.

654. To find the expansion for tan ̄1x.

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 0 < 1,

2i0 = log, (1 + i tan 6)

=

.. 0 = nπ + tan 0 – 1 tan3 0 + 1 tan3 0

But the series after na is equal to the value of between – 1π and by Art. 645.

tinuity, since tan 0 = 0;

Or, immediately by the principle of conwe have

.5
x - x3 + x5 - ...

- 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)2+qi to the form A + Bi.

a + bi = r (cos 0 + i sin 0), so that r2 = a2 + b2 and tan 0 = b/a. Then

(a+bi)p+qi = pp+qi ̧ ̧(p+qi)0i – pP ̧

=r

(q log r+pė) i

{cos (q log r + p0) + i sin (q log r +p0)}.

cos (q log r+pe) 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 e-99 has an infinite number of values; and the expression cos (q log r+pė) + i sin (q log r+pė) has values equal in number to the denominator of p.

е

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 +
0+x sin 0 + x2 sin 20 + x3 sin 30 +

S

C+ Si=1+x (cos ✪ + i sin 0) + x2 (cos 20 + i sin 20) + ...

30i

+

1

1 - x cos 0+ ix sin

1 -x (cos + i sin 0) (1 - x cos 0)2 — i2x2 sin2 0

To expand log, (1 - 2x cos + x2) in powers of x.

Log (1-2x cos 0+x2)=log.(1−xeTM – xe '+x2)=log. (1— -xe

θέλ

xe

- log, (1 − 2x cos 0 + x2) = x cos + x2 cos 20 + 3x3 cos 30 + ...