Question

设f(x)在(负无穷大,正无穷大)上有定义,且有常数T,B>0,满足f(x+T)=Bf(x),证明f(x)=a^x*g(

Answer 1

f(x+T)=Bf(x),

令B=c^T,

h(x)=f(x)/c^x,

h(x+T)=f(x+T)/c^(x+T)=Bf(x)/c^(x+T)=f(x)/c^x=h(x)

即h(x)是以T为周期的函数,

令g(x)=h(x),c=a,有f(x)=a^x*g(x)