论周期函数的导函数与原函数的周期性

On the Periodicity of the Derivative Function and the Primitive Function of a Periodic Function

研究周期函数的导函数与原函数的周期性.得到了可导的周期函数的导函数是周期函数;若f(x)是周期为T的连续函数,则f(x)的原函数F(x)是周期为T的函数的充分必要条件是∫T0f(x)d x=0;若f(x)是周期为T的连续函数,则一阶线性微分方程y'+ky=f(x)存在以T为周期的周期解的充分必要条件是,存在常数c,使等式∫0-Tektf(t)d t+c(e-kT-1)=0成立.

The periodicity of the derivative function and the primitive function of a periodic function was studied. The conclusions are as follows. At first, the derivative function of a differential periodic function is a periodic function. Secondly, let f（x） is continuous periodic function with period T, as the primitive function of f（x）, F （x） is a periodic function with period T. if and only if ∫0^Tf（x）dx=0. And finally, let f（x） is continuous periodic function with period T, linear first- order differential equation y′＋ky=f（x） has a periodic solution with period T, if and only if there is a eonstant c, suehthat ∫-T^0e^ktf（t）dt＋c（e^-kT-1）=0.