122 - Fourier Series

The Fourier Series, also referred to as a Fourier Wave Packet, is a series that is used to approximate ,within varying degrees of precision, any oscillating wave part of a periodic function, as a combination of sine and cosine waves.

Let be a half-period for the function, the time it takes for the function to complete one crest or trench, the function is then periodic in intervals of , otherwise referred to as -periodic.

The Fourier Series for the function is defined

where , and are the Fourier Series Terms, defined as

! More convenient formula for periodic functions

Most trigonometric functions are -periodic, and if that is the case, we may use a more convenient version of the series, given as

! Key point to remember

In the fourier series terms, is always zero if . Similarly, is always zero if . This can be remembered in the following: