INFINITESIMAL CALCULUS 49 by s n , we shall have log T(x + 1 ) = - yx + ^. 2 x- where y represents the limit of when fj. is indefinitely increased. This limit, whose importanc was first noticed by Euler (Ada Pctropolis, 1734), is now usually called Eulcr s Constant. If we change the sign of x in the preceding equation it becomes
fr F(l ,v) = yX + rS X~ -
Hence we have logr(l + a:) = lo Again, by logarithms, Consequently )=Jlg~ ;-jlop = * log ^irl^r " * log 1^1 + "i - ^ - ^ where Cl = 1 - 7, c 3 = -J (* s - 1 ), c s = i(* 5 - 1 ) ... It is easily seen that the constants c 3 , c 3 , &c. , form a rapidly de creasing series, in which each torm can be calculated to any required number of places of decimals. Accordingly, when the value of Kuler s constant 7 has been determined, a series of values of log F(l +x) can be computed from the foregoing equation, and thus tabulated. (q / y ) = -^T > the value of y may be calculated by making x = % in the preceding formula ; by this means its value is found to be 57721 56649 to ten decimal places. On the Integrals Lix, Six, Six, and Cix. 160. Having thus arrived at a determination of Euler s constant, we shall return to the consideration of the logarithmic integral and other transcendents introduced into 131. Adopting the notation of trfat article, we have f x e ~ 2 /~^c ~ x>l - Li(e -*)=/ dz = / du, writing xu for s ; ^c ^ U xn /! />-*- i -du + / -du. But ( 142), - I-*-. du . Again But -du. f- j y If now we suppose y to increase beyond limit, observing that in that case Lic-y=0, and that 7 = limit of 1 + i-M + +--loev y b J when y = co, we get /-I 1 _ /,-! /-I -,, /I Li(e-*) = y + log.r-/ rftt-Iim./ - !^-<fa yo ./o w We next proceed to show that y M vamshes when y becomes infinitely great. To prove this, we observe that, since u lies between and 1, Also hence 1 - (1 - u^f** <i-(i_ u ")y < yv .-. e-y-(l-u)y<yu-e-"y. -!- Consequently / /o 7" 1 1 Again, y / ue~"du = (1 -c-) ~ c - = when // O 2/ vanishes at the same time. Hence Li(c - *) = 7 + log x - f -- ^0 Again 1 , &c . Again, when e vanishes. But r 4 f ~/f Let z = xu, and this becomes -du- hence This and the preceding can be represented by the single formula Eix - Li(c) = 7 + 1 log (.r*) + a + ~ + J -^ + &c, 1 . ^ 1 U The expansion for the sine-integral can be readily obtained, for we have by definition hence, substituting the ordinary expansion for sin z, and integrat ing between the limits proposed, we get X 3 .7J 5 > " aj ~* 1.8.8 + * 1.2.3.4.5~ &C< Again, if, in the equation already proved we substitute ix for x, it becomes cos xu - % sin , , ., . log (ar) - ix - &c. Hence, equating the real parts on both sides, we get /~ l cos^cu^ .,,,,, v? . , x 4 J& u !onsequently The several scries here arrived at are readily seen to be con- ergent for all real values of x, and by aid of them the values of Cix, Six, &x for different values of the argument x can be tabulated. Such tables have been constructed by Soldner, Bidone, Bret- XIII. 7
