Question

若函数f(x)=f1(x)+f2(x),f1(x)和f2(x)分别有最小正周期T1和T2且T1/T2为有理数,则函数f(

Answer 1

T1/T2为有理数,则T1/T2可表达为m/n,m,n为正整数,mT2 = nT1=T

f1(x)是最小正周期T1的周期函数,f1(x+T1) = f1(x+nT1) = f1(x)

f2(x)是最小正周期T2的周期函数,f2(x+T2) = f2(x+mT2) = f2(x)

f(x+T) = f1(x+T) + f2(x+T) = f1(x + nT1) + f2(x + mT2) = f1(x) + f2(x) = f(x)

函数f(x)也为周期函数