omitting F(0), because it disappears in the definite integral. 118.] Other series useful for the present purpose may also be derived from Taylor's Theorem. From (84), Art. 74, Vol. I, we have the right-hand member of which rapidly converges, if X, — Xis a small quantity; and if Xr — X, is infinitesimal, taking two terms, we have which is in fact the same result as (225); and if only the first term is taken, the result is the same as (226). Again, suppose the range X--X, to be finite, and to be divided into n parts, each of which is equal to i, so that X, — Xo = ni; then Xo+2i Xo+(n-1)i and replacing the definite integrals by their values in (236), we have () dx = i {f'() + F"(xo+i) + ... + F'(xo+(n-1);)} + {"(xs)+"(xo+i) + ... + '"(+(n-1))}; (237) a nearer approximation may of course be made by including more terms of (235). 119.] We come now to a method of approximate integration, which may be very largely, and indeed almost universally applied. It is called the method of integration by series. And although it is generally inexpedient to expand a function into series, yet as this may be a case where all other methods fail, we must take the expansion as the best possible solution, and examine it as closely as we can. The process suggests itself at once. Let the element-function be expanded into a series of terms, such that each term may be integrated separately. Let these integrations be effected; hereby a new series is formed, which is the value of the given integral. If the sum of this series can be expressed in general and finite terms, that sum is the value of the definite integral; but when this cannot be found, an equality will exist between the definite integral and the series, which is often of considerable use, as to the definite integrals and the series. It must be borne in mind, that if a series adequately expresses a function, the series must be convergent: it is necessary therefore that the series to which the definite integral is equal should be convergent: this will always be the case if the series, into which the element-function is expanded, is convergent for all values of the variable within the range of integration; and may be so, even when that series is not convergent for all these values. Let f'(X) be the element-function; and let us suppose F'(x) to be capable of expansion into a series of the form Uo+uz+ Ug + ... +Um +..., which is convergent for all values of x within the range of integration; and let us suppose Rm to be the sum of all the terms after the (m+1)th, so that, the series being convergent, Rn becomes smaller the greater m is, and ultimately is infinitesimal when m = 00; then we have f'(x) = Uo+uz +Uz + ...... + Um + Rome (238) | *'(x) dx = 1*««dx +*dæ+... ...+*umdx+ /*"rdx. (239) Now, by (224), /** R. dx = (xn—Xs) x some value of R.m intermediate to those corresponding to X, and X.; but since Rom becomes infinitesimal, when m becomes infinite, so will also its mean value ; and therefore **Rm dx becomes infinitesimal and must be neglected. Hence 120.] If the element-function is capable of expansion by Maclaurin's Theorem, then the theorem given in (239) takes the following form. Since va) 1.2.3 ... rude - dx. (242) When the series (241) is convergent, the last term in (242) is infinitesimal, and must be neglected; in this case 121.] A similar process may be conveniently adopted when the element-function is the product of two functions, one of which may be expanded into a convergent series by Maclaurin's or some other equivalent Theorem. Thus suppose f'(x) = f(x) $'(x); and that f(a) = f(0) +f*(0) * +"(0) 1674 +... ... +fr-10) 1.2.3 ) +f"(0x) 1.2.3 ... : PRICE, VOL. II. If the series for f (x) is convergent, the last term of this equation must be omitted ; and we have 1.2 1x0 in which the original definite integral is expressed in a series of other definite integrals of a more simple form. The success of the method indeed depends on the possibility of integrating the several terms in the right-hand member. The following is an example of the process. Now the general term of this series is Comemory, where m is even: and by (13), Art. 82, if m is even, 1 xMdx _(m-1) (m—3)...3.17. Jo (1 – x2) m(m—2) ... 4.2 2 so that substituting these in the right-hand member we have 1 dx 1-x2)(1 - e222)}! which is a rapidly converging series, if e is a small quantity. As many examples of this process will occur in the sequel, it is unnecessary to add others here. SECTION 7.-On the Gamma-Function, and allied Definite Integrals. 122.] One of the most important definite integrals both in itself on account of the many peculiar properties which it exhibits, and in its application and the definite integrals allied to it, is le-378-1 dx. As the limits of this integral are co and 0, it is evidently a function of n only; and the symbol r(n), devised by Legendre, has been of late ordinarily employed to denote it; so that we have r(n) = ["e-*x*–1dx. (247) For this reason and for the sake of a distinctive name, the definite integral has been called the Gamma-function. It has also been called by Legendre the second Eulerian Integral, because the properties of it were originally investigated by Euler. I propose in the present section to develope these properties in an analysis of the function, so far as they fall within the scope of a general elementary treatise on the Integral Calculus. The definition of the Gamma-function is the definite integral given in (247), and the properties of it will, at first at least, be deduced from this integral. n, which is the subject-variable, is called the argument of the function. The following are other and equivalent forms of the Gammafunction, and are derived from (247) by transformation. so that the definite integrals given in (248) and (249) are definitions of the Gamma-function, as well as (247). Poisson investigated the properties of it in the form (247), and Legendre in the form (248). The following is also evident, and is a theorem of great importance ; r(n). e-axx*-1 dx = (250) |