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Fourier Transform

Source lecture(s): PHY621 Lec3

Intuition

When a function is not periodic, the Fourier series becomes an integral with continuous frequencies.

Formal Definition

\[\hat{f}(k)=\int_{-\infty}^\infty f(x)\,e^{-ikx}\,dx$$ $$f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty \hat{f}(k)\,e^{ikx}\,dk\]

Mathematical Formulation

\[\hat{f}(k)=\int_{-\infty}^\infty f(x)\,e^{-ikx}\,dx$$ $$f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty \hat{f}(k)\,e^{ikx}\,dk\]

Derivation

Let discrete spacing \(\Delta k=\pi/L\to dk/2\pi\) as \(L\to\infty\); the sum becomes an integral.

Worked Example

The Gaussian \(f(x)=e^{-x^2/2}\) transforms to \(\hat{f}(k)=\sqrt{2\pi}\,e^{-k^2/2}\).

Common Mistakes

  • Mixing up \(1/2\pi\) placement in forward vs inverse transform.
  • Confusing \(e^{-ikx}\) vs \(e^{ikx}\) sign convention.

Quiz

Q1. What does the uncertainty principle say in Fourier language?

Answer

A function and its transform cannot both be arbitrarily localized.