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3

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)

(3.1)

What happens when we stretch this interval without limit, letting ...

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