⑩天掌 Teaching Plan on Advanced Mathematics o f(x)dx 2 (2)求an f(x) cos ndx= o cos nrdc +ela. cos kx cos ndx +bs sin k cos ndx] tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 2 , 2 0 = a = − a f (x)dx 1 0 dx a kxdx b kxdx a k k k k cos sin 2 1 1 0 − = − = − = + + (2) . n 求a = − − nxdx a f x nxdx cos 2 ( )cos 0 [ cos cos sin cos ] 1 − − = + a kx nxdx + bk kx nxdx k k
⑩天掌 Teaching Plan on Advanced Mathematics o a. cos2md=an冗 f∫(x) cos nro(n=1,2,3,…) (3)求bn ∫(x) sinned sinned +Ela[ cos kx sin ndxb sin kr sin/xx l =b,T, f(x)simn(n=1,2,3,…) T tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics = − an nxdx 2 cos = , an = − a f x nxdx n ( )cos 1 (n = 1,2,3, ) (3) . 求bn = − bn f (x)sinnxdx 1 (n = 1,2,3, ) − − = nxdx a f x nxdx sin 2 ( )sin 0 [ cos sin sin sin ] 1 − − = + a kx nxdx + bk kx nxdx k k = , bn
⑩天掌 Teaching Plan on Advanced Mathematics o 傅里叶系数 f∫(x) cos ndx,(n=0,1,2,…) f(x) sin ndx,(n=1,2,… T ∫(x) cos ndx,(n=0,2,) 或 兀0 ∫(x) sinned,(n tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics = = = = − − ( )sin , ( 1,2, ) 1 ( )cos , ( 0,1,2, ) 1 b f x nxdx n a f x nxdx n n n = = = = 2 0 2 0 ( )sin , ( 1,2, ) 1 ( )cos , ( 0,1,2, ) 1 b f x nxdx n a f x nxdx n n n 或 傅里叶系数
⑩天掌 Teaching Plan on Advanced Mathematics o 傅里叶级数 +∑(anC0sHx+ b sinn) H=1 问题: f(x)条件?0+∑ (a, cos nx+ b sin nx) tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 傅里叶级数 + + =1 0 ( cos sin ) 2 n an nx bn nx a 问题: + + =1 0 ( cos sin ) 2 ( ) ? n an nx bn nx a f x 条件
⑩天掌 Teaching Plan on Advanced Mathematics o 2.狄利克雷( Dirichlet)充分条件收敛定理) 设∫(x)是以2π为周期的周期函数如果它满足条件:在 个周期内连续或只有有限个第一类间断点,并且至多只有 有限个极值点则f(x)的傅里叶级数收敛,并且 (1)当x是f(x)的连续点时级数收敛于f(x) (2)当x是f(x)的间断点时收敛于 f(x-0)+f(x+0) 2 (3)当x为端点x=±7时收敛于(兀+0)+/(=0 2 tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 2.狄利克雷(Dirichlet)充分条件(收敛定理) 设 f (x)是以2 为周期的周期函数.如果它满足条件:在 一个周期内连续或只有有限个第一类间断点,并且至多只有 有限个极值点,则 f (x)的傅里叶级数收敛,并且 (1) 当x 是 f (x)的连续点时,级数收敛于f (x) ; (2)当x是 f ( x)的间断点时,收敛于 2 f (x − 0) + f (x + 0) ; (3) 当x 为端点x = 时,收敛于 2 f (− + 0) + f ( − 0)