348 VAN orstrand: exponential function tables 



was given by Schulze and repeated by Glaisher. This value I 

 also computed by reciprocation and found the error to be 1 unit 

 in the 44th decimal. Ten interpolations between the limits x = 

 ± 0.1 and X = ±1.0 thus suffice to check the constant factors 

 e'^^-^ and the successive powers e^'^-^, e^^-^, . . . 6=^° -^ Proceed- 

 ing in this way, the functions were first evaluated at intervals 

 of unity by repeated multiplications by the factors e^^ " and with 

 these values as a basis, the values at intervals of 0.1 and 0.5 

 were obtained by successive multiplications by the factors e^^-^ 

 and e^°^ a check being obtained on the fifth and tenth interpo- 

 lations. Further independent checks were obtained by use of 

 the factors, e^^^.o, g^j-^^j e=^io.o q^}^g maximum difference between 

 any value obtained by use of the factors e^^-^ as compared with 

 the value obtained by use of the factors e^^-^ and e*^° was about 

 15 units in the 35th decimal or significant figure. Comparisons 

 based on the other factors just mentioned showed differences of 

 the same order of magnitude. Another check consisted in com- 

 puting a few isolated values by means of equation (1). 



The applications of the exponential function in pure and applied 

 mathematics are so numerous and many of them so well known 

 that it would be useless to attempt a discussion of them in this 

 paper. It appears desirable, however, to call attention to a 

 simple check which this function provides for the evaluation of 

 sin X and cos x. We may evidently write 



e"" = sin X + cos x -\- 2 



Since all of the quantities in the right-hand member of the above 

 equation are known, the value of 6^ is readily obtained by a 

 simple summation. I have applied this check to my values" of 

 sin X and cos x for x = 0.1, 0.2, . . .1.6. No errors were 

 discovered. 



In the following table the values from x = 0.0 to x = 32.0 

 have been tabulated to either 20 decimals or 20 (sometimes more) 

 significant figures. The tabular error is always <5 units in the 

 next succeeding tabular value. The more extended values are to 



6 This Journal 2 : 299. 1912. 



