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(58)

cos-1x = -sin-1x

1 203 1.3 205

= 2 X-23 2.4 5 .... And if in this equation x is replaced by ,

1 1 1 1 1.3 1 sectX = 2-2-2 3.73 – 2.4 5

(59)

195.] The preceding examples are sufficient to illustrate the process which is the subject of the present section. The equivalences given in them may again be the subjects of definite integration, and thereby will other series be found, and their sums will be found; these latter however will be in the form of definite integrals; and consequently the sums of the series will be expressed as determinate quantities only when these integrals can be evaluated. The following are examples of this process.

Ex. 1. Let the two members of (55) be the element-functions of a definite integral whose limits are x and 0; so that

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31 1/1 1 1 1 1.3 1 1 12 = 1- 2 + lž+2 3.4 22 + 2.4. 5.6 24 + ... si 100) which is a series sufficiently converging for the calculation of 7. The terms which are contained in (63) give the true value to four places of decimals.

Ex. 2. Let both members of (55) be multiplied by x, and

195.)
BY INTEGRATION.

257 then made the subjects of definite integration with the limits X and 0, we have

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Ex. 3. Again, let both members of (55) be multiplied by - dx

z; and let the definite integral be taken of both members (1 – 221' “ for the limits 1 and 0; then Ai dx P S 1 23 1.3 26 l dx JO (1-x2) And applying to each term of the right-hand member the theorem contained in (14), Art. 82, we have

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= 12 + 22 + 32 + + ...; (66) and thus the sum of the series in the right-hand member is expressed as the definite integral given in the left-hand member. But the sum of the series in the right-hand member = (139), Art. 89, Vol. I; consequently

(67) (68)

(68) 12+ 32 + 53 - 72 +.... PRICE, VOL. II.

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Section 4.—On Periodic Series, and on Fourier's Integral.

196.] The series, which, as the equivalents of certain functions, have been made the subjects of definite integration in the preceding section, have been derived from these functions primarily by the binomial theorem; and thus are perhaps the most simple of their kind. Series however formed by other processes of development may be the subjects of integration; and we intend in this section to consider one of the most important of these; viz., periodic series. The subject is important; for it exhibits new and extraordinary results which are given by no other process; and in the application of mathematical analysis to physical problems affords the solution of many intricate and curious questions, and gives expression to certain very peculiar laws. An instance of these has been adduced in Art. 15, Vol. I. in illustration of discontinuous functions; for the sum of the series therein quoted vanishes for all values of the subject-variable o, except when some multiple of n is substituted for a, when the sum takes an indeterminate form; thus the sum of the series varies discontinuously, although each term varies continuously. In this section these and similar properties of periodic functions and series will be investigated and traced to their origin.

A periodic series is that whose terms contain sines, or cosines, of the subject-variable, and of multiples of that variable. Thus A1cosa?+A2cos2a?+ ... + A„cosnx+...

is a periodic series. Whatever the sum of it may be, that sum goes through a succession of values as x increases from 0 to 2ir; because every term of the series has at the end of that period the same value which it had at the beginning; and this succession is repeated as x increases from 2 -n to 4tt; from 4w to 6 w; and 60 on; so that whatever n is, 2ir is the period of the function.

Now the primary problem is to determine the conditions for which a given function, say/(.r), is capable of expression in the

form of a periodic series, and to express it in that form. Thus, if f(x) = 4, + A, COS X + A, cos 2x + ... + A, COS N X + ....

+ B, sin x + B, sin 2 x + ... +B, sin nx + ..., (70) the problem is the determination of the unknown constants, the A's and the B's.

It will be observed that in the right-hand member of the equi. valence a non-periodic term A, is introduced; the most general form has hereby been given to the assumed equivalent of f (x); so that if f() is capable of expansion in periodic terms only, Ap = 0; the result will shew whether this is the case or not.

To determine Ao, let both members of (70) be the elementfunctions of a definite integral whose limits are a +27 and a, when a is undetermined; then since *a+2,

a+2a.
sin ix dx = 0, cosi x dx = 0;

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Thus Ap = 0, or the series equivalent to f(x) consists of periodic terms only, when / "F(x)dx = 0.

Again, let both the members of (70) be multiplied firstly by cos ix dx, and secondly by sin ix dx ; and in each case let the definite integrals of both members be taken for the same limits as before; then since &+2 Ca+27.

Ca+2a. cosic cosjc d = cosia sine da = sinx sinh da=0; (72)

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and consequently, if in these formulæ i is successively replaced by 1, 2, 3, ... n, ... , the a's and the B's will be determined. Let these be substituted in (70); and, to give greater distinctness to

the expression, let x, the subject-variable under the symbol of integration be replaced by z; then

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where the symbol 2n=1 denotes the sum of all the definite integrals in the quantity following it, which are obtained by replacing n successively by 1, 2, 3, ... 00. Thus the form and the value of the coefficients in the equivalent for f(x) which is given in (70) are determined.

In the preceding process it has been assumed that f(x) is capable of development in the given form; the complete solution however of the problem requires that the conditions requisite for the possibility of this development should be ascertained. It is also assumed that the right-hand member of (70) is equivalent to f (x); but when this is the case, the series must be convergent; and accordingly it is necessary to shew that the series is convergent in form, and to assign the limits of value of the several terms within which it is convergent. Now the theory of convergence and the tests of convergence, which have been investigated in the first section of the present Chapter, are sufficient for the purpose ; and if they are applied it will be at once seen that the required characteristics are satisfied, and that consequently an equivalence is established between the two members of (78).

197.] We need not however pursue the subject in this direction; for another course of investigation, leading to a more general result, is open to us; and I shall assume the preceding series to be only a suggestion of the form of the required de

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