Question

F(X)为奇函数,且F(X+2)=-F(X),证明其周期性

Answer 1

F((X+2)+2)=-F(X+2) = F(X) = F((X-2)+2) = -F(X-2)

对于任何X都有

F(X+4)=-F(X+2)=F(X)=-F(X-2)

所以F(X)是以4为周期的周期函数,且每半个周期变为相反数