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设f1(x)和f2(x)都是集合M上的周期函数,T1、T2分别是它们的一个周期,若T1T2∈Q,则它们的和差与积商也是M上的周期函数,T1与T2的公倍数为它们的一个周期.证明设TT12=mn(m,n∈N),记T=mT1=nT2,则T为T1和T2的公倍数且对任意x∈M,有f1(x+T)+f2(x+T)=f1(x+mT1)+f2(x+nT2)=f1(x)+f2(x)
Let f1(x) and f2(x) be periodic functions on the set M. T1 and T2 are their respective cycles. If T1T2∈Q, their sum and difference product quotient is also a periodic function on M, T1. The common multiple of T2 is their one cycle. Proof Let TT12 = mn (m, n ∈ N), remember T = mT1 = nT2, then T is the common multiple of T1 and T2 and for any x ∈ M, there is f1 (x + T)+f2(x+T)=f1(x+mT1)+f2(x+nT2)=f1(x)+f2(x)