Editing Fourier transform (section)

=== Differentiation ===
Suppose {{math|''f''(''x'')}} is an absolutely continuous differentiable function, and both {{math|''f''}} and its derivative {{math|''f′''}} are integrable. Then the Fourier transform of the derivative is given by
<math display="block">\widehat{f'\,}(\xi) = \mathcal{F}\left\{ \frac{d}{dx} f(x)\right\} = i 2\pi \xi\hat{f}(\xi).</math>
More generally, the Fourier transformation of the {{mvar|n}}th derivative {{math|''f''{{isup|(''n'')}}}} is given by
<math display="block">\widehat{f^{(n)}}(\xi) = \mathcal{F}\left\{ \frac{d^n}{dx^n} f(x) \right\} = (i 2\pi \xi)^n\hat{f}(\xi).</math>

Analogously, <math>\mathcal{F}\left\{ \frac{d^n}{d\xi^n} \hat{f}(\xi)\right\} = (i 2\pi x)^n f(x)</math>, so <math>\mathcal{F}\left\{ x^n f(x)\right\} = \left(\frac{i}{2\pi}\right)^n \frac{d^n}{d\xi^n} \hat{f}(\xi).</math>

By applying the Fourier transform and using these formulas, some [[ordinary differential equation]]s can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "{{math|''f''(''x'')}} is smooth [[if and only if]] {{math|''f̂''(''ξ'')}} quickly falls to 0 for {{math|{{abs|''ξ''}} → ∞}}." By using the analogous rules for the inverse Fourier transform, one can also say "{{math|''f''(''x'')}} quickly falls to 0 for {{math|{{abs|''x''}} → ∞}} if and only if {{math|''f̂''(''ξ'')}} is smooth."