EVALUATION OF CERTAIN DEFINITE INTEGRALS. 147 



Let us suppose z > m ; then the lower of each pair of am- 

 biguous signs must be taken, and the expression within the 

 brackets may be written thus 



(m+z)"- 1 - ~ (m+z-2)-*+& c . +^ ( m ^ z -2) n - l ^(m-z) n '\..(q). 



As, from the nature of the case, m and n are either both 

 odd or both even, if m is even, n - I is odd, and therefore 

 (m - z} n ~ l = - (z m} n ~ l 'j and thus, in every case, (#) equals 



(1 - D~T (* + m) n ~ 1 ... {D$z = <t>(z + 2) say,} 

 and this is ^ m D~^ n (z + m} n ~ l = A w (z - m) n ~ l = 0, 

 since mis>nl. Consequently 



sin wl x , 



cos zx ax 0, when z is > m, 



i. 



a remark not made by Laplace; when m = n = l 9 its truth is 

 known. 



As (q) = 0, add it, multiplied by f - -^ ^ , to the value 



\n IJ 2 



of the integral already found, therefore 



where the series stops whenever the next term would intro- 

 duce a negative quantity raised to the power n 1. This is 

 easily seen to be true, for every such term will have a different 

 sign in (q) 9 and in the definite integral, and thus on addition, 

 all such terms will disappear. Equation (4) is Laplace's form ; 

 the discontinuity of the function is now expressed, not by 

 ambiguous signs but by the stopping short of the series at 

 different points. 



By a similar method, we find that 



... (5), 



which might have been deduced from (4) by integrating both 

 sides without introducing any complementary quantities. This 

 remark is general; having once established the general form 

 of (4) for any given value of w, we may deduce from it that 

 which corresponds to any other value of w, simply by differ- 

 entiation or by integration, without bringing in any constants. 



102 



