Introduction to Fourier Series
Basically, Fourier series express a function as a sum of sine or cosine terms. They are useful because many differential equations are linear, so that if
and
are solutions to the equation then so is
for suitable
and
and by extension, any linear combination of solutions is also a solutions, with suitable coefficient
Expressing a function as a sum of sine and cosine terms is especially useful because many wavefunctions are naturally expressed in terms of linear combinations of sine and cosine terms.
Any continuous function defined on an interval can be expressed as a Fourier series, and any function with a finite number of discontinuities. The function is extended outside the range on which it is defined by defining it as a repeating function, because sin and cosine are repeating functions with period
so for example, if
Then we can extend
outside the interval
by writing
where
This is illustrated below.
In general for
We can extendoutside the interval by writing
We can in fact extend a function in more than one way, besides the sample given above. We can extendto be an even function, so that
or we can extendto be an odd function, so that
while maintaining periodicity.
For these two extensions the period has doubled, but periodicity remains.