CHAPTER VI COMPUTATION OF LOGARITHMS AND OF THE TRIGONOMETRIC FUNCTIONS-DE MOIVRE'S THEOREM -HYPERBOLIC FUNCTIONS 57. A convenient method of calculating logarithms and the trigonometric functions is to use infinite series. In works on the Differential Calculus it is shown that where e 2.7182818... is the base of the Naperian system = of logarithms. 58. The series (1) converges only for values of x which satisfy the inequality I<≤1. The series (2), (3), and (4) converge for all finite values of x. It is to be noted that the logarithm in (1) is the Naperian, and the angle x in (2) and (3) is expressed in circular measure. COMPUTATION OF LOGARITHMS 59. We first recall from Algebra the definition and some of the principal theorems of logarithms. The logarithm to the base a of the number m is the number x which satisfies the equation, This is written x = loga m. ax = m. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. Thus loga mn= loga m+logan. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor. The logarithm of the power of a number is equal to the logarithm of the number multiplied by the exponent. To obtain the logarithm of a number to any base a from its Naperian logarithm, we have 60. We proceed now to the computation of logarithms. The series (1) enables us to compute directly the Naperian logarithms of positive numbers not greater than 2. Example. To compute loge to five places of decimals. 3 If the result is to be correct to five places of decimals, we must take enough terms so that the remainder shall not affect the fifth decimal place. Now we know by Algebra that in a series of which the terms are each less in numerical value than the preceding, and are also alternately positive and negative, the remainder is less in numerical value than its first term. Hence we need to take enough terms to know that the first term neglected would not affect the fifth place. Subtracting the sum of the negative from the sum of the positive terms, we Denote the sum of the remaining terms of the series by R. Then, by Algebra, Therefore The error caused by retaining no more decimal places in the computation is less than .0000006. Hence the total error is less than .0000027. the result is correct to five decimal places. 61. As remarked, the series (1) does not enable us to calculate directly the logarithms of numbers greater than 2, but it can be readily transformed into a series which gives us the logarithm of any positive number. which converges for -1<<1. Putting y=(1+x), passes from 3 5 4 207 7 we see that y passes from o to ∞ as x I to +1; hence, if we make this substitution in which converges for all positive values of y, and therefore enables us to compute the Naperian logarithm of any number. From (5) we can get another series which is useful: put which converges for all positive values of y. Hence, equation (5) gives us I I I + This series gives us log. (y+1), when log, y is known. It converges more rapidly than (6), when y is greater than 2, and hence should be used under these circumstances. 62. To construct a table we need to compute directly only the logarithms of prime numbers, since the others can be obtained by the relation log xy=log x+log y. |