[MILLER A rosebrugh] VALUES OF FUNCTIONS INVOLVING e-x 75 



subject to the limitation that the seriesin the exponent of e must converge 

 rapidly. 



Of the integrals 



f-Y- dx, f — dx, I e-^ dx, fxe-^- dx, fx'^e-'^ dx, 



The first may be reduced to the second, the so-called " Exponential 

 Integral " Ei {-x), and the fourth and fifth to the third, or Exponential 

 Function. 



The Exponential Function has been tabulated by Mr. F. W. New- 

 man,^ from .r = to X = 37, at intervals of .1 to eighteen decimal 

 places (sixteen exact), from a; =: to j: = 15.349 at intervals of . 001 

 to twelve decimal places, from x = 15.350 to a: = 17.298 at intervals of 

 .002, and from x = 17.300 to a: = 27.635 at intervals of .005 to 

 fourteen decimal places. 



Tables of the same function have been prepared by Mr. J. W. L- 

 Grlaisher^ from a; = . 001 to :c = 1 . COO at intervals of . 001, from a: = . 01 

 to X =: 2.00 at intervals of .01, from x = 0.1 to a; = 10.0 at intervals 

 of .1, and from x = 1 to .x = 500 at intervals of unity. In every case 

 the fii-st nine significant figures are given. 



The Exponential Integral has been tabulated by Mr. J. W. L. Glaisher^ 

 from X = to X = 1.00 at intervals of .01 to eighteen places, from 1.0 

 to 5.0 at intervals of . 1 to eleven places, and from 6.0 to 15.0 at inter- 

 vals of unity to eleven places. 



For the problems with which we were concerned, these intervals were 

 too wide, we have therefore constructed a table of the descending Expo- 

 nential Integral from x = to x' = 1.000 at intervals of .001, and from 

 X = 1.00 to X = 2.00 at intervals of .01, to nine decimal places. 



The other three integrals have not hitherto been tabulated, so far as 

 we are aware. 



00 



Valuis of /(?-■« dx, from x = to x — l.OwO at intervals of .001 



X 



{Table I), and from x = 1.00 to x = 2.00 at intervals of .01 {Table IL). 

 These are Newman's values of e-"^ to the nearest digit in the ninth decimal 

 place. 



In his "Tables of the Exponential Function" G^laisher says : " fhe 

 last figure is therefore in general correctly given to the nearest unit, but 

 it may be in error by a unit where the tenth figure is a 4, 5 or 6." On 

 comparing Grlaishei-'s table with that of Newman five cases were found 



1 Cambridge Phil. Trans., XIII, 145 (1883). 



2 Ibid, XIII, 24.S. 



3 Phil. Trans., CLX, 367 (1870). 



