⑩天掌 Teaching Plan on Advanced Mathematics o 例3设f(x)是以2为周期的连续函数,且 ∫(x)=+∑(anc0smx+ b sinn)可逐项积分, H=1 2 试证明:f2(x)=+∑(an+bn2), 其中an,bn为f(x)傅立叶系数 证∵!f(x)=“0+∑( a cosn+ b sinn) f(x)=f(x)+2Ianf(x)cos nx+b,f(x)sin nxI tianjin polytechnic dmivendityTianjin Polytechnic University Teaching Plan on Advanced Mathematics 设f (x)是以2为周期的连续函数，且   = = + + 1 0 ( cos sin ) 2 ( ) n an nx bn nx a f x 可逐项积分， 试证明： ( ) , 2 ( ) 1 1 2 2 2 2 0    =  − = + +  n an bn a f x dx 其中a , b 为f (x)的傅立叶系数. n n 证   = = + + 1 0 ( cos sin ) 2 ( ) n an nx bn nx a  f x   =  = + + 1 2 0 ( ) [ ( )cos ( )sin ] 2 ( ) n f x an f x nx bn f x nx a f x 例 3