TABLES OF THE EXPONENTIAL FUNCTION AND OF THE CIRCULAE SINE 

 AND COSINE TO EADIAN ARGUMENT.' 



By C. E. Van Ohstrand. 



The tables accompanying this paper have been prepared with the expectation of meeting 

 a twofold requirement. The first was to obtain a few high place values at sufficiently small 

 intervals of argument for general use in the evaluation of integrals and other functions; the 

 other object was to obtain a basis for subsequent interpolation to small intervals of argument 

 for use in the construction of complete 10-place tables which are applicable in the various 

 fields of pure and applied mathematics. The need of tables meeting these and other require- 

 ments has been emphasized by various authors. 



The most important tables of extended values of the exponential function in which the 

 exponents are integers or fractions have been constructed by Schulze, Bretschneider, Newman, 

 Gram, Glaisher, and Burgess. Bretschneider included a few high place values of the circular 

 sine and cosine to radian argument, but with the exception of these and a few values computed 

 by Gudermann, there appears to be no extended values of these important functions. 



Schulze ' gives values of the ascending exponential at intervals of unity between the limits 



1 and 24, inclusive, to 28 or 29 significant figures, and for the special arguments 25, 30, and 60 



his values include 32 or 33 figures. In so far as I have been able to ascertain, Schulze gives no 



information regarding methods of computation or accuracy of results. Glaisher ^ verified the 



first 15 figures of Schulze's value of e'" by direct substitution in the series; the first 13 powers of e 



were verified to 22 places of decimals; and the values of e", e", ... e" to 15 places of decimals 



by means of the relation 



e"— 1 



7=e''"' + e""' ... +e + l. 



e— 1 



Bretschneider * evaluated e, c"', sin 1 and cos 1 each to 105 places of decimals; also values 

 of the same functions at intervals of unity between the limits 1 and 10, inclusive, to 20 places 

 of decimals. He corrected the erroneous value of e given in Callet's tables and Vega's Thesaurus 

 and the slightly erroneous values of sin 1 and cos 1 given to the twenty-fifth decimal by 

 Gudermann." 



Bretschneider obtained his values by direct substitution in the exponential series in con- 

 nection with the evaluation of the three transcendants, 



> Published by permission of the Director of the U. S. Geological Survey. 



' I. S. Schulze, Sanunlung logarithmischer Trigonometrischer-Tafeln (Berlin, 1778). 



• J. W. L. Qlaisher, Tables of theerponential function. Camb. Phil. Trans., vol. 13 (1883), pp. 243-272. (In Salomon's Tafeln (1827) the Tsluei 

 of «n, e-n «"•"", ... £«J<»«»n where n has the values 1, 2, ... 9 are given to 12 places. 



* C. A. Bretschneider. Berechnung der Grundzahl der natiirlichen Logarithmen, so wie mehrerer anderer mit ihr Eusanunenbangender Zahl- 

 werthe. Grunert's Archiv der Math, und Phys., Bd. 3 (1S43). pp. 27-34. 



> C. Qudennann. Potential oder cyklsch-hyperbolische Functionen. Jour, ftjr die reine imd angewandte Math., Bd. VI (1830), pp. 1-89. 



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