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Mathematics for Physics

348 Fourier Transform

An integral transform of a function that is used to express the function through continuous frequency components.

Note: Also written \(\hat{f}\). Also written \(\tilde{f}\). Also called the Fourier transform of \(f\).

definition [d] (Fourier Transform) From Olver: the Fourier transform of a function \(f(x)\) is

  • \(\displaystyle \hat{f}(k) = \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)\, e^{-ikx}\, dx\) .

The inverse Fourier transform recovers the original function by

  • \(\displaystyle f(x)\sim\dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \hat{f}(k)\, e^{ikx}\, dk\) .

where

  • \(f\) is the original function.
  • \(\hat{f}\) is the Fourier transform of \(f\).
  • \(x\) is the original variable.
  • \(k\) is the frequency variable.
  • \(i\) is the imaginary unit.

definition [d] (Fourier Transform) From Riley, Hobson, and Bence: the Fourier transform of \(f(t)\) is

  • \(\displaystyle \tilde{f}(\omega) = \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(t)\, e^{-i\omega t}\, dt\) ,

and its inverse is

  • \(\displaystyle f(t) = \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \tilde{f}(\omega)\, e^{i\omega t}\, d\omega\) .

where

  • \(f\) is the original function of \(t\).
  • \(\tilde{f}\) is the Fourier transform of \(f\).
  • \(t\) is the original variable.
  • \(\omega\) is the frequency variable.
  • the product of the two normalizing constants equals \(\dfrac{1}{2\pi}\).

Forward transform. The transform integrates \(f\) against a complex exponential of frequency \(k\). This principle is used to assign a complex amplitude to each frequency.

Inverse transform. The inverse integral rebuilds \(f\) from those amplitudes. This principle is used to recover the original function from its Fourier transform.

Symmetric normalization. Placing \(\dfrac{1}{\sqrt{2\pi}}\) in both formulas keeps the pair as symmetric as possible. This principle is used to fix the constant factors in the transform pair.

348.1 Examples

348.1.1 Simple

Take the right-handed exponential pulse \(f_{r}(x)=e^{-ax}\) for \(x>0\) and \(f_{r}(x)=0\) for \(x<0\), with \(a>0\). Its Fourier transform is

\[ \hat{f}_{r}(k) = \dfrac{1}{\sqrt{2\pi}\, (a+ik)} \]

where

  • \(a\) is a positive decay rate.
  • \(k\) is the frequency variable.
  • \(\hat{f}_{r}\) is the Fourier transform of \(f_{r}\).

348.1.2 General

For \(f(t)=0\) when \(t<0\) and \(f(t)=A e^{-\lambda t}\) when \(t\geq 0\), with \(\lambda>0\), the transform is

\[ \tilde{f}(\omega) = \dfrac{A}{\sqrt{2\pi}\, (\lambda+i\omega)} \]

where

  • \(A\) is the amplitude at \(t=0\).
  • \(\lambda\) is a positive decay rate.
  • \(\omega\) is the frequency variable.
  • \(\tilde{f}\) is the Fourier transform of \(f\).

348.2 References

  1. Olver, P. J. Introduction to Partial Differential Equations. Springer, 2016. Chapter 7, §7.1, pp. 264–267 — Fourier transform \(\hat{f}(k)\); inverse formula; exponential pulse example.
  2. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. 3rd ed. Cambridge University Press, 2006. Chapter 13, §13.1, pp. 434–435 — symmetric Fourier transform pair; exponential decay example.
  3. Cahill, K. Physical Mathematics. Cambridge University Press, 2019. Chapter 4, §4.1–4.2 — Fourier transform pair; Gaussian transform.
  4. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 5th ed. Matrix Editions, 2015. §4.11, Definition 4.11.24 — Fourier transform \(\hat{f}(\xi)=\int_{\mathbb{R}} f(x)e^{2\pi i\xi x}\, dx\).