Page:Philosophical Transactions - Volume 145.djvu/186

There are some other definite integrals which we may use in the summation of factorial series, as

${\begin{aligned}&\int _{0}^{\frac {\pi }{2}}d\theta \cos ^{n}\theta \cos n\theta \cos 2r\theta ={\frac {\pi }{4}}\cdot {\frac {n(n-1)\ldots (n-r+1)}{1.2.3\ldots r}}{\frac {1}{2^{n}}},\\&\int _{0}^{\pi }{\frac {d\theta \sin ^{2n}\theta }{(1-2a\cos \theta +a^{2})}}={\frac {(2n-1)(2n-3)\ldots 3.1}{2n(2n-2)\ldots 4.2}}\cdot {\frac {\pi }{2}},\\&\int _{0}^{1}{\frac {dx\ x^{\alpha -1}(1-x)^{\beta -1}}{(x+a)^{\alpha +\beta }}}={\frac {1}{a^{\beta }(1+a)^{\alpha }}}\cdot {\frac {\Gamma \alpha \Gamma \beta }{\Gamma (\alpha +\beta )}},\\&\int _{-\infty }^{\infty }{\frac {dx}{(a+ix)^{m}(b-ix)^{n}}}=2\pi (a+b)^{1-m-n}\cdot {\frac {1.2.3\ldots m+n-2}{1.2.3\ldots m-1.1.2.3\ldots n-1}},\\&\int _{0}^{\infty }{\frac {x^{m}-{\frac {1}{2}}dx}{\{(x+a)(x+b\}^{i}}}={\frac {\Gamma {\frac {1}{2}}\Gamma \left(n-{\frac {1}{2}}\right)}{\Gamma n}}{\frac {1}{({\sqrt {a}}+{\sqrt {b}})^{2n-1}}},\end{aligned}}$

and probably some besides.

I shall now offer a few observations on the nature of the integrals we have been discussing. The preceding investigations appear to be equivalent to a solution of the following problem:—"To find the definite integrals, whose values can be determined in finite terms by the solution of linear differential equations with variable coefficients. "It should seem that the definite integrals, which we have considered in this paper, are the most general ones of any importance, whose values can be found in this way, for the following reasons:—If we expand any definite integral, which is a solution of a differential equation, and its equivalent in terms of the principal variable, and equate like powers of that variable, we obtain a series of definite integrals of a simpler kind, each equal to a fraction whose numerator and denominator consist of factorials, and can therefore be expressed by the products of Eulerian integrals, or to the sum of such fractions. Now I have employed all the more important definite integrals of this class, which are yet known, in the summation of the series which satisfy the differential equation

$(ax^{n}+bx^{n-r}){\frac {d^{n}y}{dx^{n}}}+(a'x^{n-1}+b'x^{n-r-1}){\frac {d^{n-1}y}{dx^{n-1}}}+\mathrm {\&c.} =0;$

and as the properties of the Eulerian integrals have been much studied, and the integrals whose values are dependent on them consequently well known, it is probable that the definite integrals, which we have considered in this paper, embrace all the more important ones whose values can be determined in finite terms by the solution of the above equation. Were we to employ equations of a more general form, we should find that the successive terms of the series which express their solutions, would be given by equations of finite differences, in which the members equated to zero would each consist of more than two terms. Consequently we should be unable in the general case to sum the resulting series by means of definite integrals; and in those cases in which we might find this possible, the integration of the differential