The Fourier transform of a function forms a new function of a particular type. The transform is useful because it can be made to produce a series of periodic functions equivalent to the original function (in some sense like a Taylor series), if the original function is periodic (or limited to a finite domain). This is commonly used for the analysis of periodic functions representing waves such as electromagnetic radiation or sound. As such, it is typically used to determine what sine-like components make up a particular periodic function, i.e., break a signal down into its harmonics. A variant on the transform (inverse Fourier transform or backward Fourier transform as opposed to the normal, or forward Fourier transform) can "undo" this. Despite being opposite transformations, the computation work to carry them out is virtually the same and software that does one can easily be used to do the other.
The forward transformation is:
F(s) is ∞ ∫ f(x)e-2πisxdx -∞
The inverse transformation produces the original function, f(x), from this F(s).
The fast Fourier transform algorithm, an efficient means of computing a type of Fourier transform is ubiquitous in physics, engineering analysis, and modeling.