f(x)={πe^x 0

题目

f(x)={πe^x 0

答案

由f(-x) = f(x),f(x)为偶函数,可知其Fourier级数不含正弦项.
对正整数n,∫{-π,π} f(x)cos(nx)dx = 2π·∫{0,π} e^x·cos(nx)dx.
e^x·cos(nx)的一个原函数为e^x·(cos(nx)+n·sin(nx))/(1+n²).
于是∫{-π,π} f(x)cos(nx)dx = 2π·(e^π·cos(nπ)-1)/(1+n²) = 2π·(e^π·(-1)^n-1)/(1+n²).
于是对正整数n,cos(nx)的系数为2(e^π·(-1)^n-1)/(1+n²).
常数项 = ∫{-π,π} f(x)dx/(2π) = ∫{0,π} e^xdx = e^π-1.
f(x)的Fourier级数为e^π-1+2·∑{1 ≤ n} (e^π·(-1)^n-1)·cos(nx)/(1+n²).