 cos(kw0t) cos(gw0t)dt 0  (5) 0 where 2f (6) w0  1  (7) f T where and represents the frequency (in cycles/time) and period (in seconds) respectively. Also, f T and are integers. k g A periodic function with a period should satisfy the following equation (t ) f T (t ) (t) T  f f (8) Example 1 Prove that  sin(kw0  0 t) 0 for 2f  w0 1 f  T and is an integer. k 0  Solution  0 Let T  t)dt A 0     1 T    cos(kw t) 0 0 kw0 (9)    1     A   0   (10)  1   cos(k2)1
 cos(kw0t) cos(gw0t)dt  0 (5) 0 where w0  2f (6) f  1 (7) T where f and T represents the frequency (in cycles/time) and period (in seconds) respectively. Also, k and g are integers. A periodic function f (t ) with a period T should satisfy the following equation f (t T )  f (t) (8) Example 1 Prove that  sin(kw0 t)  0 0 for w0  2f f  1 T and k is an integer. Solution Let T A  t)dt 0    1  T  cos(kw t)   0 0  kw0   1  A      0   1    cos(k2)1 0   0 (9) (10)
U4-L18-_-Fourier-Series-and-examples.docx