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

(1+4)

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(1)

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. When x = 1 we have 1

2
1
1+ =1+1+ +

to...

() 2!

3!

(1-7(1-2)

n

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+

X2 that is,

et = 1 + x +

+

2! 3! In particular, if x = 1 we have

1 1
e=1+1+ + +

2! 3!

We cherefore see that we can compute the value of e 11.000000 by simply adding 1, 1, 1 of 1, $ of 1 of 1, and so on,

2 1.000000

3 0.500000 indefinitely, and that to compute the value to only a few

4 0.166667 decimal places is a very simple matter. We have merely 50.041667 to proceed as here shown.

6 0.008333 Here we take 1, 1, 1 of 1, $ of 4 of 1, 4 of } of f of 1, 7 0.001388

8 0.000198 and so on, and add them. The result given is correct

90.000025 to five decimal places. The result to ten decimal places

0.000003 is 2.7182818284.

e = 2.71828.

i sin x =

ܕ6:

COS X

COS X =

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

cos 1 + i sin 1=k. Then

cos x + i sin x =(cos 1 + i sin 1)* = kt, and, putting

a for x,
cos (– 2) + i sin ( - 2) = cos

= k-x. That is,

cos x + i sin x = and

i sin x = k-3. By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2 i, we have

1
2

(K* + k--),

1 and

sin x =

(k* k-r).
But km =(e logk)* = ex logą, and k-* = e-x logk.
er logk = 1 + x log k +

** (log k)? _ ** (log k)
+

+
2!

3!

x2 (log k) 28 (log k) and e-clogk = 1- x log k +

+ 2!

3! 1

** (log k), ** (log k) .. COS X = (k® + k-x)=1+

+

+ 2

2! 1

moo (log k) 8 206 (log k) and x log k +

+

+ 3!

5! Dividing the last equation by x, we have

1

x2 (log k) ** (log k)' log k +

+

+ 3!

5!

2 i

4!

sin x =

{

...}.

sin x

X

log k.

But remembering that a 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 a becomes, the wearer comes to 1, and 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

1 = log k; whence

log k = i, and

kref.

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From the last two series we obtain, by division,

2c8 2206 1777
tan x= = x + + +
COS X

3 15 315

sin a

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 57.3o, we have

1 1 1 1 cos1=1

+

+
2! 4! 6! 8!

=1-0.5 + 0.04167 — 0.00139 + 0.00002

= 0.5403 = cos 57° 18'.

+

COS X =

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,

23 25 wc? sin x = x

+ 3! 5! 7!

хв and

1 +

+

2! 4! 6! By multiplying by i in the formula for sin x, we have

ix8 irs ix? i sin x = ix

+

+

3! 5! 7! Adding,

ix8

ixo cos x + i sin x=1+ ix

+ +

2! 3! 4! 5! By substituting ix for x in the formula for ex, we see that

2222 2828

సం
eix = 1+ ix + + + + +

2! 3! 4! 5!

X2 ix: 24 ixo =1+ ix

+ +

2! 3! 4! 5! In other words,

elx = COS X + i sin x.

X2

x2

or

e

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.

Sin = cos x + i sin x, in which a may have any values, we may let x = 7. We then have

ett = COS TT + i sin =- =-1+0,

ein =-1. In this formula we have combined four of the most interesting numbers of mathematics, e(the natural base), i (the imaginary unit, V-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 ein =-1, we have

TE V-1=i. Taking the logarithm of each side of the equation ein =-1, we have

in = 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 0 = cos (2 km + $)+ i sin (2 km + o), we see that ex, which is equal to cos $ + i sin ø, may be written e(2 kn + ), or we nay write

= cos $ + i sin = cos (2 km + $)+ i sin (2 km + ) Hence (2 km + ) i = log[cos (2 km + )+ i sin (2 km + )]. If = 0,

2 Επί =

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

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

If $ =, (2 km + T)i=(2k + 1) i = log (-1).

Hence the logarithms of negative numbers are always imaginary; for if k = 0 we have ni = log (-1); if k=1 we have 3 ni = log (-1); and so on. If we wish to consider the logarithm of some number N, we have

Ne2 kmi = N(cos 2 km + i sin 2 kt).
Hence log N + 2 kri = log N + log (cos 2 km + i sin 2 km)

log N + log 1 = log N. That is, log N = log N + 2 kni. Hence the logarithm of a number is the logarithm given by the tables, + 2 kmi. If k = 0 we have the usual logarithm, but for other values of k we have imaginaries.

epi

= e(2 km + Øli

Exercise 83. Properties of Logarithms Prove the following properties of logarithms as given in $ 159, using b as the base :

1. Properties 1 and 2. 3. Property 4. 5. Property 6. 2. Property 3.

4. Property 5. 6. Property 7 Find the value of each of the followiny : 7. 5! 8. 7! 9. 6! 10. 8!

11. 10! Simplify the following : 10! 10! 7! 15!

20! 12. 13. 14. 15.

16.
3!
8!
5!
14!

17! 1 1

+ + 2! 3!

1

1 1 18. Find to five decimal places the value of (2 + + +

2! 3! By the use of the series for co8 x find the following: 19. cost. 20. cos } 21. cos 2.

22. cos 0.

2

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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: 27. tan 0. 28. tan 1.

29. tant.

30. tan 2. Prove the following statements : 31. eant = 1.

32. e

= .

33. e* = V-1. 34. e= V1.

TT

2

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

35. 2.

36. 4.

37. V2.

38.

- 2.

Given log 5 = 1.609, find three logarithms (to the base e) of: 39. 5. 40. 25. 41. 125.

42. 5.

46. V10.

Given log 10 = 2.302585, find two logarithms (to the base e) of : 43. 100.

44. 10. 45. 1000. 47. From the series of 162 show that sin(-6)=- sin 0.

48. Prove that the ratio of the circumference of a circle to the diameter equals – 2 log ()=-2 i log i.

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