we When we are able to effect the integration of any function, the determination of its value between certain limits of the independent variable offers in gencral no difficulty, as have merely to subtract its value at one limit from its value at another. There are however many functions, the Definite Integrals of which we are able to find, although the indefinite integral cannot be expressed in finite terms. The evaluation of these integrals has become one of the most important branches of the Integral Calculus, in consequence of the numerous applications which are made of them both in pure mathematics and in physics : it is to functions of this kind that the examples in the following paper refer. The methods for evaluating those definite integrals whose general values cannot be found are very various, but they can generally be classed under the following heads. (1) Expansion of the function into series, integration of each term separately, and summation of the result. (2) Differentiation and integration with respect to some quantity not affected by the original sign of integration. (3) Integration by parts of a known definite integral, so as to obtain a relation between it and an unknown one. (1) Multiplication of several definite integrals together, so as to obtain a multiple integral, and, by a change of the variables in this, converting it into another multiple integral, coinciding with the first at the limits, and admitting of integration. By this means a relation is found between the definite integrals multiplied together, which frequently enables us to discover their values (5) Conversion of the function by means of impossible quantities into a form admitting of integration. These different methods will be best understood by their application to the following examples. We shall begin with the function known as the Second Eulerian Integral, because, though its exact value cannot be found generally, its properties have been much studied, and to it a number of other integrals are reduced. 1. Second Eulerian Integral. The definite integral L* dx e-*20-', when n is a whole number, is easily seen by the method of reduction in Ex. (13), Chap. 11. of the Integ. Calc. to be (n-1)... 3.2.1. When, however, n is a fraction, its value can be found only in certain cases, but it possesses many remarkable properties which render it of the greatest importance in the Theory of Definite Integrals. It was first studied by Euler, who seems at an early period to have seen its importance, and has devoted several memoirs to the investigation of its properties ; on this account Legendre has named it after him, at once for the purposes of characterizing the function and honouring that great mathematician. To distinguish it from another integral with which also Euler had much occupied himself, and of which we shall afterwards treat, it is usually called the “Second Eulerian Integral," and Legendre has affixed to it the characteristic symbol I, applied to the index, so that he writes So* dx e-*4"-1 = (n), which notation we shall adopt. Throughout the following investigations n is supposed to be greater than 0. In the first place we remark that by a change of the independent variable this integral may be put under other forms which are sometimes more convenient in practice than that which we have used. Thus if we put e-s= y, the corresponding limits are x = 0, y = 1; x = 00 , y = 0, This is the shape under which the integral has been usually treated both by Euler and Lagrange, but it is scarcely so convenient as the preceding. Again, if we put @" = x, the limits remain the same as before, and we have (6) r n 6 If n = This last form is the most convenient for determining the value of the integral in one remarkable case when it can be found in finite terms. = } 6* dx e-* x-! = r (1) = 2 * dž€–39. Let k = f*dze: then as the value of the definite integral is independent of the variable, we have also k = 6* dy e-", and therefore multiplying these together, kʻ = f* dže-*.6" dy e-* = 6* %* dy dx e-(42+z?) since y and < are independent. Now assume x = r cos , y = y sin 0, then dydx = rdr de. To determine the limits we observe that y and % never become negative, and therefore 0 must vary from 0 to while no yaries from 0 to 00 , so that we have k* = * ** dr der e-** = 1; whence (c) k = }} and T (5) = 7%. We shall now demonstrate the more important properties of the function 1 (n) referring the reader who wishes, for a more detailed exposition of them to Legendre, Exercices de Calcul Intégral, Tom. I. and il. If we integrate by parts the expression sdx e-*w" we have Sdx e-*w" = - € *x" + n sdx e-*x*-1. The integrated part vanishes at both limits, so that (d) r(n + 1) = n r(n). This may be looked on as a characteristic property of the function I, and is of the greatest importance, as by means of it we can reduce the calculation of r (n) from the case when n>1 to that when it is < 1, and we have therefore to occupy ourselves only with the values of n which lie between 0 and 1. If n be a proper fraction, and therefore r(n) r (1 – n) = f*dx e-* x*-? Som dy e-"y-* 60* %* dx dy c-(2+4) xn-y-. To reduce this, we shall use the transformation of Jacobi, given in Chap. III. Sec. 2, Ex. (7) of the Diff. Calc. Assume x + y = U, y = uv, so that dæ dy = u du du: the limits of u and u corresponding to those of w and y, are u=0, U = 0, v = 0, v = 1; therefore r(n) 1 (1 – n) = 6* S'du dv e-ky-" (1 – v)"); or, integrating with respect to u between its limits, r(n) 1 (1 – n) = foydu v-" (1 – v). To find the value of this integral, assume v = (sin 6)"; then, as to the limits x = 0, x = 1, correspond 0 = 0, 0 = {T, , we have r (n) (1 – n) = 2 **d 0 (tan 6).–21. Now tan 0 = ( - ) - and it is therefore obvious that (tan 0)1-9" may be expanded into a series of the form 2n-1 2 c.}: 1-2 n 2 (-) 2 {1 + A, 6-(-)+20 + A, 6-(-)* ** + &c.} 1- (-) (A, sin 20 + A, sin 40+ &c.) = cos (1 – 2n). + (-)} sin (1 – 2n) sin n 7 + (-) cos nm. Substituting the series for (tan 6)1-2", multiplying by sin nr + (-) cos n it, and equating real parts, we have sin na I (1) (1 - 1) = 2 %d0(1 + A, cos 20 + A, cos 40 + &c.) 2 TT =; since the periodic terms vanish at both limits. Hence It is easily seen that the value of r() is found at once from this equation, by putting n = }; and generally, if we know the value of l (n) from n = 1 to n = we know its value from 0 to From the preceding theorem a more general one may be derived. Let n be a positive integer, then will $ I' Г r = (27)"="n-4. n ist. Let n be even : Then there are I pairs of factors of the form 1 (1) 1 (1 – ), and a middle factor which is r (?) = 74. Therefore by the preceding theorem the product is equal to 11 - 1 * This demonstration of a Theorem discovered by Euler is given by Mr Greatheed in the Camb. Math. Journal, Vol. 1. p. 17. |