46 INFINITESIMAL CALCULUS 1.2. . ~ u}"F (u)du This is Lagrange s series, in which the remainder after n terms is exhibited in the form Discontinuous Integrals. 152. The integral calculus furnishes many examples of discon tinuous functions. For example ^ sin ax cos bx , _ t /" sin (a + &)a; , _ , f x sin (n - b),r , _ - "~ ft < =* YT / C(/tK- ~r o / CtX a; yo # l/o x When a + b and a-b are both positive, each of the latter integrals (148) is equal to - . Hence we have- when a > b, and when a < b , x sin ax cos bx , ir aX =s ^r ) x 2 " sin ax cos foe , n If Z>, the value of the integral becomes Here we have an example of a function of two variables a and b, changing its value suddenly when b, varying in a continuous manner, becomes equal to or greater than a. This singularity has been in geniously iitilized for the purpose of obtaining the values of certain definite integrals. For example, let then, since u = Q when a is less than b, and u = ~ when a>b, we A have /oo /- ue~ a da = I 2/6 or But 2jb ax cos "V6 T e /CO jv, e~ a sin ax da , ( 134) ; 1 + x* /* cos bx T+^ C Again, considering b as variable, 6 sin ax cos foe 77 udb= Jo Jo Hence, if b < a, we have dxdb=t sin ax sin bx / sin ax
sin ax sin bx if b>a, we have Consequently/"* S Jo smbx dx ig al to 5_ multiplied by the a; 2 2 smaller of the numbers a and b. Again, let us consider the definite integral (a-b cos x}dx /*_^ /o 2 - Here we have (a b cos x}dx lab cos x + b* /"J y a 2 -: COS X + 6 2 = _ a 2 - ~2 2 Again (Ex. 8, 113), f J ,1 /Yi + 5? 2a J a 2 - 2ab b*r dx J a 2 - 2ab cos x + b z cos x + b 2 dx _ 2 - lab cos x + b 2 = a 2 - dx . t^-Wf laj a 2 - 2ab cos Accordingly, if a- > & 3 , 0_J rir fa % a Jo <$ - lab cos If a*<b, we liavc p tan-i 1 a + b , y T tan a-b 2 J tan-M"- 1 -; tan ; a a-b X _- ) 2 / 2a a* - 5 2 /" _dx _j7r^ 2a A a- - 2ab cos x + 6 2 2a Consequently, when a 2 - b~ > 0,
- "" (a-b cos x)dx
when a- - V 1 < 0, /" r (a-b cos x) / and when a = Z>, Jo 2<t J (1 - cos a;) 2_/ The comparison of these three cases shows that if b be supposed to vary in a continuous manner from a value less than a to a value greater than a, the integral /"" (a-b cos x)dx a* - 2ab cos x + 6 2 will assume for b = a-h, a, a + h the values JL JL 0. It is a 2 accordingly a discontinuous function. Eulerian Integrals. 153. The following definite integrals, /Vi were first studied, under a modified form, by Euler, who devoted several memoirs to the investigation of their properties. They were afterwards fully discussed by Legendre, by whom they were styled Eulerian integrals of the first and second species respectively. The latter integral is now regarded as the fundamental one, to which the other is reducible, as shall be presently shown. In the case where n is an integer we plainly have /-co / c-*x n - l dx=I.2.3 . . . n= . Jo The integral is in all cases a function of n ; and, when n is frac tional, it is regarded as a distinct transcendental function. It was distinguished by the symbol T by Legendre, thus : This is now usually called the Gamma-Function, but sometimes, however, the Factorial Function, a name suggested by Arbogast, and subsequently adopted by Kramp and others. Moreover, since (Ex. 6, 115), we have F(n + 1) = ?ir(?i). This may be taken as the fundamental property of gamma- functions, and by aid of it the calculation of all such functions is reduced to the case where the parameter n is comprised between any two consecutive integers. The values of r( 0> or rather of log T(n), were tabulated to twelve decimal places by Legendre in his Traite desfonctions elliptiques, tome 2, ch. 16, corresponding to values of n increasing by intervals of 001 between the integers 1 and 2. It may be remarked that T(l) = 1, T(0) = 00, T(-n) --=<*>, n being an integer. For negative values of n, not being integers, the function has a finite value. The first Eulerian integral, I x m - l (l-x) n - l dx, is evidently a function of its two parameters, m and n. Following Binet we shall represent the integral by the notation B(m, n). It is readily seen that l /-l r I +f Its value, when either m or n is a positive integer, can be immedi ately found. For, suppose n a positive integer, then ( 123) we have By successive applications we get ) ..... 1_ _ / 1 a . w -,^. - 2) ... (wi + 1 )/ Q . (? + - 1.2.3. .. (n-l) + 2) . . . (m + n-l)
