FOURIER'S THEOREM 93 



latter an even function of x*. This is illustrated by the 

 annexed graphical representations, in which f(x) is given 

 primarily only for the range TT, but is continued in one 

 case as an odd and in the other as an even periodic function 

 of x. It will be noticed that in the former case the stipulation 

 that f(x) is to vanish for x = and x = TT is necessary if 

 discontinuities are to be avoided. 



Since any function f(x) given arbitrarily for values of x 

 ranging (say) from TT to TT can be resolved into the sum of an 

 even and an odd function, viz. 



/(*) = i !/(*)+/(-*)} +i {/(*)-/<- *)). -<i) 



we derive the more general theorem that the sum 



f m (x) A + A! cos x + A z cos 2# + . . . + A m cos mx 



4- B! sin x + B 2 sin 2x + ... + B m sin mx, ...(2) 

 where 



* An "odd" function is one which is simply reversed in sign with x, 

 like x 3 or sinx. An "even" function is one which is unaltered in value 

 when the sign of x is changed, like x- or cos x. 



