Fourier analysis MOC

Fourier transform

The Fourier transform is a unitary operator on of that converts a function from a time domain to a frequency domain: instead of describing amplitude of a function at a given time, the transformed describes the amplitude of a function at a given frequency.

1 The Fourier transform may be thought of as a complex version of the Laplace transform, or an extension of the Fourier series from discrete integer frequencies to continuous real ones. In any case, the inverse Fourier transform is significantly simpler than the inverse Laplace transform:

Note on derivation

The above formulae can be derived from the Fourier series by taking the limit as . See 2018. Introduction to quantum mechanics, problem 2.19, p. 60

Properties

The Fourier transform has the properties23

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Applications

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Footnotes

  1. Used above is the unitary angular form. ↩

  2. 2023. Advanced Mathematical Methods, p. 92 ↩

  3. Proofs can be found in §9.5: Properties of the Fourier Transform (LibreTexts) ↩