⑩天掌 Teaching Plan on Advanced Mathematics o 第八节一般周期函数的傅里叶级数 周期为2L周期函数的傅里叶级数 二、总结 tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 第八节 一般周期函数的傅里叶级数 一、周期为2L周期函数的傅里叶级数 二、总结
⑩天掌 Teaching Plan on Advanced Mathematics o 周期为2L周期函数的傅里叶级数 2兀 T=2l,0= 代入傅氏级数中 +>(a, cos nax +b sin nax) 2 定理设周期为的周期函数f(x)满足收敛 定理的条件则它的傅里叶级数展开式为 ∫(x0+∑(a11 ntx nTtr +b, sin -), 2 tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 周期为2L周期函数的傅里叶级数 T = 2l, . 2 T l = = 定理 定理的条件则它的傅里叶级数展开式 为 设周期为 的周期函数 满足收敛 , 2l f (x) ( cos sin ), 2 ( ) 1 0 l n x b l n x a a f x n n n + = + = ( cos sin ) 2 1 0 a n x b n x a n n + n + = 代入傅氏级数中
⑩天掌 Teaching Plan on Advanced Mathematics o 其中系数an,b为 nTtr f(x)cos",x,(n=0,1,2,…) nth f(x)sin",dx,(n=1,2,…) (1)如果f(x)为奇函数,则有 f(x)=∑bsin nTcr 其中系数b为b2 T x sin dx (n=1,2,…) tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 其中系数an , bn为 ( )cos , ( 0,1,2, ) 1 = = − dx n l n x f x l a l l n ( )sin , ( 1,2, ) 1 = = − dx n l n x f x l b l l n (1)如果f (x)为奇函数, 则有 ( ) sin , 1 = = n n l n x f x b ( )sin , 2 0 dx l n x f x l b b l n n 其中系数 为 = (n = 1,2, )
⑩天掌 Teaching Plan on Advanced Mathematics o (2)如果(x)为偶函数则有 f(x)="+∑ a. cOS nTo n=1 其中系数a为n-=2 nTx x)cOS (n=0,,2,…) 证明△,_πx ,一l≤x≤l→-兀≤z≤π 设f(x)=f()=F(z),F(z)以2m为周期 T F(3)=+2(an cos nz+b, sin nz), tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics (2)如果f (x)为偶函数, 则有 cos , 2 ( ) 1 0 = = + n n l n x a a f x dx l n x f x l a a l n n = 0 ( )cos 2 其中系数 为 (n = 0,1,2, ) 证明 , l x z 令 = − l x l − z , ( ) ( ) F(z), lz f x f = 设 = F(z)以2为周期. ( cos sin ), 2 ( ) 1 0 a nz b nz a F z n n = + n + =
⑩天掌 Teaching Plan on Advanced Mathematics o 其中 "F( z)coS naz 几=兀 z) SIn nza。 Tr z= F(=f(r) n f(x)=+2(an cosx+b, sin x) 2 其中an1=,,f(x)cos,xcr, n ∫(x)sin"xx tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics ( cos sin ) 2 ( ) 1 0 x l n x b l n a a f x n n n + = + = ( )sin . 1 ( )cos , 1 − − = = b F z nzdz a F z nzdz n 其中 n ( )sin . 1 ( )cos , 1 − − = = l l n l l n xdx l n f x l b xdx l n f x l 其中 a F(z) f (x) l x z = =