Fourier and Laplace transforms
The complex exponentials exp(i2πnx/L) are orthonormal and easy to differentiate (and to integrate), but they are periodic with period L. If one wants to represent functions that are not periodic, then a better choice is the complex exponentials exp(ikx), where k is an arbitrary real number. These orthonormal functions are the basis of the Fourier transform. The choice of complex k leads to the transforms of Laplace, Mellin, and Bromwich.
3.1 The Fourier transform
The interval [–L/2, L/2] is arbitrary in the Fourier series pair (2.37)
What happens when we stretch this interval without limit, letting ...