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161. Exponential Series. By the binomial theorem we may expand

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This is true for all values of x and n, provided n > 1. If n is not greater than 1 the series is not convergent; that is, the sum approaches no definite limit. The further discussion of convergency belongs to the domain of algebra.

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| 1.000000 21.000000

3 0.500000

We therefore see that we can compute the value of e by simply adding 1, 1, † of 1, § of † of 1, and so on, indefinitely, and that to compute the value to only a few decimal places is a very simple matter. We have merely 5 0.041667 to proceed as here shown.

Here we take 1, 1, † of 1, † of 1⁄2 of 1, 1 of } of { of 1, and so on, and add them. The result given is correct to five decimal places. The result to ten decimal places is 2.7182818284.

4 0.166667

6 0.008333

7 0.001388 8 0.000198 9 0.000025 0.000003

e = 2.71828.

162. Expansion of sin x, cos x, and tan x. Denote one radian by 1,

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By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2i, we have

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But remembering that x represents radians, it is evident that the smaller x is, the nearer sin x comes to equaling x; that is, the more nearly the sine equals the arc.

Therefore the smaller x becomes, the nearer

sin x

comes to 1, and

x

the nearer the second member of the equation comes to log k. We therefore say that, as x approaches the limit 0, the limits of these two members are equal, and

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From the last two series we obtain, by division,

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By the aid of these series, which rapidly converge, the trigonometric functions of any angle are readily calculated.

In the computation it must be remembered that x is the circular measure of the given angle.

Thus to compute cos 1, that is, the cosine of 1 radian or cos 57.29578°, or approximately cos 57.3°, we have

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163. Euler's Formula. An important formula discovered in the eighteenth century by the Swiss mathematician Euler will now be considered. We have, as in § 162,

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By multiplying by i in the formula for sin x, we have

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By substituting ix for x in the formula for e, we see that

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164. Deductions from Euler's Formula. Euler's formula is one of the most important formulas in all mathematics. From it several important deductions will now be made.

Since eix

cos x + i sin x, in which x may have any values, we may let x = π. We then have

or

e1 = cos π + i sin π = — 1 +0,
ein = 1.

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In this formula we have combined four of the most interesting numbers of mathematics, e(the natural base), i (the imaginary unit, √−1), (the ratio of the circumference to the diameter), and — 1 (the negative unit). Furthermore, we see that a real number (e) may be affected by an imaginary exponent (in) and yet have the power real (— 1).

Taking the square root of each side of the equation ei" =— we have

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- 1,

Taking the logarithm of each side of the equation ei=— 1, we have

iπ = log(-1).

Hence we see that 1 has a logarithm, but that it is an imaginary number and is, therefore, not suitable for purposes of calculation.

Since cos + i sin = cos (2 kπ + $) + i sin (2 kπ +ø), we see that e, which is equal to cos & + i sin ☀, may be written e(2 kπ+6)i, or we may write

edi = e(2 kπ + b)i = cos + i sin = cos (2 kπ + $) + i sin (2 kπ + $) Hence (2 km+)i=log[cos(2 k +¢)+isin(2 k +¢)]. If = 0,

2 kπi = log 1.

If k = 0, this reduces to 0 = log 1.

If k = 1 we have 2 Ti = log 1; if k = 2, we have 4 πi = log 1, and so on. In other words, log 1 is multiple-valued, but only one of these values is real.

If π, (2 kπ + π) i = (2 k + 1)πi = log(-1).

Hence the logarithms of negative numbers are always imaginary; for if k = 0 we have wi = log (− 1); if k = 1 we have 3 πi = log(-1); and so on.

If we wish to consider the logarithm of some number N, we have

Hence

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log N+2 ki= log N + log (cos 2 km + isin 2 km)

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That is, log N = log N + 2 kπi. Hence the logarithm of a number is the logarithm given by the tables, + 2 kπi. If k = 0 we have the usual logarithm, but for other values of k we have imaginaries.

Exercise 83. Properties of Logarithms

Prove the following properties of logarithms as given in § 159,

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1 1

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2! 3!

17. Find to five decimal places the value of (1+1+ +

18. Find to five decimal places the value of (2 +

1 1
+ +
21 3!

By the use of the series for cosx find the following:

19. cos.

20. cos.

21. cos 2.

22. cos 0.

By the use of the series for sin x find the following: 23. sin 1.

24. sin .

25. sin 2.

26. sin 0.

By the use of the series for tan x find the following:

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Given log, 2 = 0.6931, find two logarithms (to the base e) of: 35. 2.

36. 4.

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Given log.5 = 1.609, find three logarithms (to the base e) of :

39. 5.

40. 25.

41. 125.

42. 5.

Given log 10 = 2.302585, find two logarithms (to the base e) of:

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47. From the series of § 162 show that sin (— 4):

46. V10.

sin .

48. Prove that the ratio of the circumference of a circle to the

diameter equals — 2 log (i) = 2 i log i.

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