= 275$ (0). (188) If a = 0, then Sof"(cev=1) dô = 2f”(O). (189) In (188) replace 0 by – 0; and a by –27, then $*$'(re-ov55)de = 2^ f'(o). (190) Thus (188) is true when 0 is replaced by -0. If the subject-variable z, given in (185), is replaced throughout by x+2, where x is a variable independent of 2, so that the element-function is f'(x+reov-1); then f'(0)=f'(x), and we have 1 panta $'() = /****F"(x+rev=ī) do ; . (191) so that any function of x, f'(x), where x is possible or impossible, may be expressed as a definite integral, of which the elementfunction is the original function, provided that f'(x+reov-1), is finite and continuous for all values of the modulus, and that the value of f'(treor -]) is not altered, when 25+ 0 is substituted for 0. Ex. 1. Let f'(z) = ; thus f'(z) is finite and continuous for all values of z less than 1: in this case then the modulus qe must be less than 1; therefore by (189), . . C2. do ... Jo 1-reov si = 27; P20 (1-re Ov=1) de *.1-r cos 0 +rV-1 sin o -do; so that equating possible and impossible parts, we have for all values of r less than 1,: do but I = 47 log 2-4 7 log 2, by (75), Art. 94; = 0. (199) 110.] This is all the account that I am able to give, consistently with the scope of the present treatise, of the method of evaluating definite integrals devised by M. Cauchy. Much more might be said both on the simple cases which have been investigated, and on the numerous applications of the resulting formulæ, as well as on definite integrals, the direct forms of the element-functions of which are more complicated. But for all these I must refer the student to the original memoirs of M. Cauchy. In the course of his studies the reader will find that when the factor, for which f'(z) = oo, is of the form (2-5) ", the correction for infinity takes a more complicated form than that given in (155); and M. Cauchy has devised for the determination of it a new process involving new symbols and a new algorithm, which he calls Calcul des Résidus. This however is one of the higher parts of the subject which I have not attempted to develope. It will be observed that almost all the definite integrals to which the method has been applied have also been deduced by other processes ; indeed Cauchy has drawn hardly any results which had not been demonstrated by other methods. The method requires very great caution : but theoretically there are scarcely any limits to the extent of application of the equation (126), which is the fundamental theorem of the method. It is the nearest approach to a general method of evaluating definite integrals that has, as yet, been discovered. It will also be observed that in the preceding Articles the subject-variable z of the definite integral has been restricted to the very particular form z = x+yV-1; although it is theoretically very general, being any function of x and y. The application of the general theorem to another form, viz. z = x(a +yV ) has been briefly and imperfectly made in Art. 104: and the correction for infinity, which is one of the most useful parts of the method, has not been made in that case. I may in conclusion remark that the process requires the determination of the roots of f'(z) = 00; so that it can be employed only when the roots can be found. SECTION 6.—Methods of Approximating to the Value of a Definite Integral. 111.] If the correct value of a definite integral cannot be determined by any of the methods explained in the preceding sections, yet there are many methods by which an approximate value of it can frequently be found; and the limits also can be determined within which the error of approximation lies. These methods will be investigated in this section. We must recur to the precise definition of a definite integral given in equations (20) and (21), Art. 83 ; and we have /*'(x) dx = f(mm) — F(x) =(4,-&)F"(x)+(x2-x1) F'(x) + ... + (x rs — Xn—2) F'(x,-1); (200) subject to the condition that f'(x) does not become infinite for any value of x within the range of integration. Now (200) is a series of terms each of which is the product of two factors. One of the factors is an element of the range of integration, and the other is a given function of the variable, the variable having a given value; thus the latter factor is a quantity completely defined; but the former factor is an element of the range, and is arbitrary, provided that it is infinitesimal, because the mode of partition of the range is arbitrary. Different modes of partition are suitable to different element-functions; but doubtless that which is most generally applicable, and which is also the most simple, is the partition of the range into equal elements. Let us adopt this mode; and accordingly let us suppose x,-, to be divided into n equal parts, each of which = i; so that X,, - Xo = ni, and X-Xo = x,—X; = ...... = X, — Xn-1 = i. (201) then (200) becomes (**»'(x) dx = i{'()+'(x.+i)+''(*+21) +...I'{+(n-1)i} }; (202) and for the evaluation of the definite integral it is necessary to find the sum of the series of functions contained in the righthand member. This sum will generally be a function of n and i; and consequently of n only, since Xn-X = ni; and if in it oo is substituted for n, so that i is an infinitesimal, the resulting value will be the value of the given definite integral. Although some examples of this mode of evaluating a definite integral have been given in Art. 9, it is desirable to add others; for the notion of a definite integral, as the sum of a series, cannot be too frequently impressed on the student, whether for the sake of an exact idea, or for the purposes to which this Calculus will be applied in both the present and the subsequent volumes of our course. peo sin x , Ssin o sin i sin 2 i sin oo dx = X -+ - + : +...+..sin i sin 2 i sin 3 i =i+ 1 + 2 + 3 +... ad infinitum + = 5, when i is infinitesimal. (203) the same result as (76), Art. 94. Ex. 2. /" log (a? – 2a cos 0 + 1)do, when a is greater than 1. Let the range be divided into n equal parts, each of which = then the definite integral (204) = 25 log a. which is the same result as (198), Art. 109. |