Expand method (condition)

Fourier series expansion of periodic functions

nature

If the function is an odd function

If the function is an even function

Function expansion defined on [-l, l]

Function expansion defined on [0, l]

Note: The function f is not defined in intervals other than the interval [0,l], so there are countless ways to extend it, and therefore there are countless kinds of expansions, but they all represent the function f on (0,l), and the function value equal.

Through conversion, the exponential form of the trigonometric function is used to generate the Fourier series (basis function) in the complex form

Fourier integral in real form

Fourier transform in real form

If the non-periodic function f(x) is an odd function

If the non-periodic function f(x) is an even function

Fourier integral theorem

Fourier integral in complex form

Fourier transform in complex form

The complex Fourier integral and Fourier transform can be converted into each other, for example

symmetrical form

The original function is an even function and the derivative is an odd function.