The motivation for the Fourier transformation comes from the study of a Fourier series. In the study of a Fourier series, complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. Due to the properties of sine and cosine, it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use Euler's formula, which states that e = cos 2 πθ + i sin 2 πθ, to write Fourier series in terms of the basic waves e, with the distinct advantage of simplifying many unwieldy formulas.