THE INTEGRATION OF PERIODIC FUNCTIONS 377 



cos (n -f w)0 cos (n - m)0~| * 



-it- 

 -a- 



M + 



n - 



cos 2(n + m)n I cos 2(n 



w -f w 



n - 







t 



sin wO cos rnO dO 



cos (w + m)0 cos (n m)0 



-i- 



I 



n+ m n m 



)TC cos{ (n + m)} TC 



n + m 

 cos (n m)7t cos { (n 



n-m)7r}"| 



= 



sin 7i0 cos w0 



-it- 



i r co 



=: 2L 



cos (n+ w)0 cos (n-- m)0 



W 



nm 



T 



-' 



cos (n + m)Tc - 1 cos (n 



nm 



n+m 

 = 0, when n + m is even, 



= 1 when n + m is odd 



7i+ m nm 



Hence the integrals Isinn0d0, Icos7i0d0, I sin n0 sin rn0 d0, 



I cos 7i0 cos 7W0 d0, and I sin n0 cos w0 d0 all vanish when taken 



between the limits and 27c, and they all vanish when taken 

 between the limits TC and TT. If and TC are taken as the limits 



of the integrals, then the integrals I cos w0 d0, I sin n0 sin m0 dft, 

 and I cos n0 cos m0 d0 vanish, while the integrals I sin n0 dQ 

 and I sin w0 cos 7n0 d0 do not vanish. 



187. These results do not apply to the case when n = m, except 

 for the integral I sin w0 cos mft dft 



