INVERSE METHOD OF DEFINITE INTEGRALS. 379 



44. Examples. 



In the following examples two things are to be observed. First, 

 that the given functions are supposed to be continuous, and therefore 

 the equation proposed must hold true for all values of the parameter a. 



Secondly, In the final equation for determining the unknown coeffi- 

 cients, instead of using a reciprocal multiplier any means more simple 

 may be occasionally employed. 



Ex. 1. Given ^^(/), cos («^) = 1 to determine <^{t\ ^he limits of t 

 being and tt. 



Put (^{t) = Co + Ci cos/ + d cos (2/) + d cos (3/), &c. ml infinitum, 



and cos {at) = Ao + A^ cos t + A2 cos (2/) + A^ cos (3/), &c., 



where to determine Ao, A^, A,., &c. we multiply successively by 1, 

 cos t, cos 2 A &c., and integrate from t = to t = ir, whence 



, _ sin {a-n) J _ 2asin«-7r , _ 2«sin«7r 



J n ^ / ,x„ 2« sin air , 

 and generally A^ = ( — 1) . — ri ^ when n> 0. 



7r {a — n J 



Multiply both series and integrate, and we get by the proposed 



equation, 



[Co a.€i , «C2 aCi . \ 



1 = sm a-K { — =- + ~„ — -„ z — -5 + &c.> 



{a a^—1 «^ — 2^ «^ — 3^ J 



Put a = 0, 1, 2, 3, &c. successively, and we get 

 _ 1 2 3 . 



C(j — , Cj — , C2 = — , oZC. 



•TT TT TT 



Hence ■tr(p{t) = 1 + 2cos/ + 2cos2# + 2cos3#, &c. 



The value of <t)(t) is therefore the transient function - . =^^ — ^-.'^ ^ — i-^ . 

 ^^ ^ TT I — Hh cos t + h^ 



{Vide Art. 38. Function Fo), when h is put equal to unity. 

 Vol. V. Part III. 3D 



