161. Exponential Series. By the binomial theorem we may expand 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. | 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, By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2i, we have 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 From the last two series we obtain, by division, 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 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, By multiplying by i in the formula for sin x, we have By substituting ix for x in the formula for e, we see that 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, 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 п - 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 log N+2 ki= log N + log (cos 2 km + isin 2 km) 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, 1 1 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 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: Given log, 2 = 0.6931, find two logarithms (to the base e) of: 35. 2. 36. 4. 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: 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. |