40 INFINITESIMAL CALCULUS In this latter shape it is called the exponential integral. Hence, adopting Schloimlch s notation, we write Jo (r , f x COS ~ f -* c-* Ei=/ - dz J-j> z Li *=/"*! J() lg ~ Again, if &-/ Jv U , we have An interesting and valuable historical account of these tran scendental functions is given by Mr J. W. L. Glaisher in the Trans actions of the Royal Society, 1870, of which want of space prevents our giving a fuller account. Mr Glaisher has also, in the same memoir, given tables of the numerical values of these transcen dental functions for a number of different arguments. It may be added that the logarithmic integral was discussed, and tabulated by Soldner in 1809. Numerous integrals have been reduced to depend on the fore going transcendents. For example, in the great tables of Bierens de Haan (Nouvelles tables d integralcs definics, Leyden, 1867) nearly 450 forms are shown to be reducible to one or other of the functions considered in this section. What has been said here will help to exhibit the way in which the necessity for the introduction of new transcendental functions arises as the calculus is developed, and to show that around each new transcendent whole classes of integrals are grouped. 132. The very limited number of differentials which can be inte grated in a finite form by aid of the ordinary functions makes it an interesting and important question to find whether the integral of any proposed differential expression is capable of being represented by such functions or not. This problem appears to have been first discussed in a general manner by Abel. Our limits admit only of a statement of one or two of the general results thus arrived at. The reader will find a tolerably full account of the treatment of the question in Bertrand s Calcul Integral, pp. 89-110. Abel s fundamental theorem may be stated as follows. Suppose y to be an algebraic function of the variable x, that is, a function defined by a rational equation (x, y) = 0, which is of the wth de gree in y ; then, if the integral fydx be also an algebraic function of x, it must be of the form fydx = P + PjT/ + IV/ + &c. + P,, _ iy - , in which P , PJ , P. 2 . . . P,,-] are rational functions of x. The functions P fl , Pj. . . . can be investigated by the method of in determinate coefficients, which, in the great majority of cases, will show the impossibility of an algebraic integral. In the particular case where y=^ ^/X, X denoting a rational func tion of x, it has been shown by Liouville, as a consequence of Abel s theorem, that, if the integral /c&e /X be algebraic, it must be of the form P! tJX., in which Pj is a rational algebraic function of x. Again, denoting X by , and substituting T for M"- 1 ^", if the integral f where M and T are whole polynomials, be expressible algebraically, it is of the form m , , where is another polynomial. v J- If the equation be differentia ted, we see that the highest degree of x in must be one greater than that in M. Hence, by the method of indeterminate coefficients the integral, if it is algebraic, can be found ; or else it can be shown to be impossible under such a form. Again, if t, u, v, . . . be algebraic functions of x, the differential of t + A log u + B log v + &c. , where A, B, C are constants, is evidently algebraic. The converse theorem was investigated by Abel, viz., when y is algebraic, to find liKiJydx can be expressed by algebraic and logarithmic functions. He showed that if fydx = t + A log u + B log v + C log w + &c. , then the functions t, it, v, . . . are capable of being expressed as integer functions of y. Abel s theorem was extended by Liouville, who started from sup posing fydX"F(x, c", e, log it; . . . ), where K, v, u~, &c., are algebraic functions of x. He proved thnt, when y is algebraic, the expression for its integral cannot contain an exponential, such as c". Also that a logarithmic function, such as log w, cannot enter into the integral except in a linear form with a constant coefficient. In particular, it is shown by Abel that whenever / is ex- pressible explicitly, it must be of the form VR + A log VR VR lo in which P and R arc integral polynomial functions of .->. Definite Integrals. 133. The investigations have thus far been chiefly limited to what are styled indefinite integrals. It is plain from 107 that, when ever the expression <f>(x) remains finite between the limits of inte gration, its definite integral, taken between those limits, can be de termined whenever its indefinite integral is known. For instance, since dx 1 + 2x cos a + J? we have f 1 dx _J_i 1 JQ 1 4- 2x cos a + x 2 sin a ( Also (Ex. 8, 113), 1 , fX + COS a . tan- 1 : sin a, sin a . 1 + cos a . , cos a) a 1 ; --tan- 1 - . sin a sin a) 2 sin a dx 1 + cos a cos x sin a ---: tan- 1 f tan tan ~ Accordingly From this we readily get 1 + cos cos x sin a dx - . when /.- 2 dx JQ a 2 cos 2 a; + b 2 sin 2 x 2ab 134. As definite integrals have frequently to be considered in which we regard one or both of the limits as infinite, it is necessary to determine whether the equation holds for infinite limits. Suppose when X becomes infinitely great that F(.r) approaches a finite limit, represented by F(w), then lim. r Y(x}dx = Y n I F(X) - F(a; ) I F(oo)-F(.r ) . X=OO/- X = C<3^ l
- / **
Consequently the formula holds in this case. In like manner if, when x becomes - oo , F(.r tends to a finite value F( - oo), we have Also /foo F(aO*B-F(oo)-F(->).
Hence, when F (o?) remains finite between the limits, and F(a ) has determinate values for both limits, the equation always holds. For example, in the integral and when x~= -co, , > a- + x 2 a when x =00, tan- 1 | ) has for its limit - ir - aj 2 tan- 1 has for limit- ; hence a 2 /-*> dx _ TT r X (7X __ TT Jo a 2 + x 2 2a J a 2 + x 2 a Also, from the integrals given in 114, we get
