Rather than Taylor series, that are supposed to work for “any” function, we shall study periodic functions. For periodic functions the French mathematician introduced a series in terms of sines and cosines,
|
f(x) = {{a}_{0}\over
2} +{ \mathop{∑
}}_{n=1}[{a}_{n}\mathop{ cos}\nolimits (nx) + {b}_{n}\mathop{ sin}\nolimits (nx)].
| (4.5) |
We shall study how and when a function can be described by a Fourier series. One of the very important differences with Taylor series is that they can be used to approximate non-continuous functions as well as continuous ones.