CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 



175 



please to the superior limit. In this case put for shortness f(9) = 5, regard |et - 9 as a func- 

 tion of $, F(S), and integrate from $ = to 5 = /3, where /3 is a constant which may be as 

 small as we please. By what precedes, F'($) will be finite in the integration, and may be 

 made as nearly as we please equal to the constant i^(0) by diminishing /3. Hence the integral 



ultimately becomes 84.F*(D)fl&£ e~ cf>s d$, which vanishes when c becomes infinite. Hence the 



limit of Ti is zero. 



We have evidently 



t 3 < r e -''dt, 



which vanishes when b becomes infinite. Hence the limit of T is equal to that of T 2 . Now 

 making c first infinite and afterwards a and b, we get 



limit of T, = limit of / 6-''dt = / e~ fi dt = ^/ v , 

 and therefore we have ultimately, for very large values of c, 



«-©'•"■ 



In order to apply this expression to the integral u given by (6) we must put 



(22) 



3q' 



I 



we 3 "', whence q = 



»)M^ S 



?Vn/T\* 



U?/ (3w)i 



_^V" 



1,293 = 2 (?) ^~ h 



whence we get ultimately 



«= " e * \*> 4 * 

 (3n)i 



Comparing with (15) we get 



U = 



(3w)i 



cos 



H?)'-;}' 



A = B = 



2*3*' 



(23) 



(24) 



9. We cannot make n pass from positive to negative through a series of real values, so 

 long as we employ the series according to descending powers, because these series become illu- 

 sory when n is small. When n is imaginary we cannot speak of the integrals which appear 

 at the right hand side of (5), because the exponential with a positive index which would appear 

 under the integral signs would render each of these integrals divergent. If however we take 

 equation (6) as the definition of u, and suppose U always derived from u by changing the sign of 

 \/ - 1 in the coefficient of the integral and in the value of p, but not in the expression for n, 

 and taking half the sum of the results, we may regard u and U as certain functions of n 

 whether n be real or imaginary. According to this definition, the series involving ascending 



* This result might also have been obtained from the in- 

 tegral U in its original shape, namely, / cos (x 3 -nx)dx, 



Jo 



by a method similar to that employed in Art. 21. If x x be the 

 positive value of x which renders x 3 -nx & minimum, we have 

 *i =3-*»*. Let the integral {/be divided into three parts, by 

 integrating separately from x = to x = x, - a, from x = x, - a 



tox = x 1 + b, and from .r = x, + 6 to x = co ; then make n infinite 

 while a and b remain finite, and lastly, let a and b vanish. In 

 this manner the second of equations (23) will be obtained, by 

 the assistance of the known formulae 



r 



J -a 



co$x 2 dx 



r 



sin x 2 dx — 2~* vK 



