Exponential Fourier series指数傅里叶级数

cite from wikipedia

Exponential Fourier series

We can use Euler's formula,

where i is the imaginary unit, to give a more concise formula:

Assuming f(x) is a periodic function with T = 2π, the Fourier coefficients are then given by:

The Fourier coefficients a_n, b_n, c_n are related via

and

The notation c_n is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of f (in this case), such as   or F, and functional notation often replaces subscripting:

In engineering, particularly when the variable x represents time, the coefficient sequence is called afrequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

 与傅里叶级数原始形式比较 

由于cosnx=cos(-nx),sin(nx)=-sin(-nx)