Editing Fourier transform (section)

=== Distributions, one-dimensional ===
The Fourier transforms in this table may be found in {{harvtxt|Erdélyi|1954}} or {{harvtxt|Kammler|2000|loc=appendix}}.
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks
|-
|
|<math> f(x)\,</math>
|<math>\begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|Definitions
|-
| 301
|<math> 1</math>
|<math> \delta(\xi)</math>
|<math> \sqrt{2\pi}\, \delta(\omega)</math>
|<math> 2\pi\delta(\omega)</math>
|The distribution {{math|''δ''(''ξ'')}} denotes the [[Dirac delta function]].
|-
| 302
|<math> \delta(x)\,</math>
|<math> 1</math>
|<math> \frac{1}{\sqrt{2\pi}}\,</math>
|<math> 1</math>
|Dual of rule 301.
|-
| 303
|<math> e^{i a x}</math>
|<math> \delta\left(\xi - \frac{a}{2\pi}\right)</math>
|<math> \sqrt{2 \pi}\, \delta(\omega - a)</math>
|<math> 2 \pi\delta(\omega - a)</math>
|This follows from 103 and 301.
|-
| 304
|<math> \cos (a x)</math>
|<math> \frac{ \delta\left(\xi - \frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2}</math>
|<math> \sqrt{2 \pi}\,\frac{\delta(\omega-a)+\delta(\omega+a)}{2}</math>
|<math> \pi\left(\delta(\omega-a)+\delta(\omega+a)\right)</math>
|This follows from rules 101 and 303 using [[Euler's formula]]:{{br}}<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math>
|-
| 305
|<math> \sin( ax)</math>
|<math> \frac{\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)}{2i}</math>
|<math> \sqrt{2 \pi}\,\frac{\delta(\omega-a)-\delta(\omega+a)}{2i}</math>
|<math> -i\pi\bigl(\delta(\omega-a)-\delta(\omega+a)\bigr)</math>
|This follows from 101 and 303 using{{br}}<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math>
|-
| 306
|<math> \cos \left( a x^2 \right) </math>
|<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) </math>
|<math> \frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>
|<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right) </math>
|This follows from 101 and 207 using{{br}}<math>\cos(a x^2) = \frac{e^{i a x^2} + e^{-i a x^2}}{2}.</math>
|-
| 307
|<math> \sin \left( a x^2 \right) </math>
|<math> - \sqrt{\frac{\pi}{a}} \sin \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) </math>
|<math> \frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>
|<math> -\sqrt{\frac{\pi}{a}}\sin \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right)</math>
|This follows from 101 and 207 using{{br}}<math>\sin(a x^2) = \frac{e^{i a x^2} - e^{-i a x^2}}{2i}.</math>
|-
|308
|<math> e^{-\pi i\alpha x^2}\,</math>
|<math> \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\pi \xi^2}{\alpha}}</math>
|<math> \frac{1}{\sqrt{2\pi \alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}}</math>
|<math> \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}}</math>	
|Here it is assumed <math>\alpha</math> is real. For the case that alpha is complex see table entry 206 above.
|-
| 309
|<math> x^n\,</math>
|<math> \left(\frac{i}{2\pi}\right)^n \delta^{(n)} (\xi)</math>
|<math> i^n \sqrt{2\pi} \delta^{(n)} (\omega)</math>
|<math> 2\pi i^n\delta^{(n)} (\omega)</math>
|Here, {{mvar|n}} is a [[natural number]] and {{math|''δ''{{isup|(''n'')}}(''ξ'')}} is the {{mvar|n}}th distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all [[polynomial]]s.
|-
| 310
|<math> \delta^{(n)}(x)</math>
|<math> (i 2\pi \xi)^n</math>
|<math> \frac{(i\omega)^n}{\sqrt{2\pi}} </math>
|<math> (i\omega)^n</math>
|Dual of rule 309. {{math|''δ''{{isup|(''n'')}}(''ξ'')}} is the {{mvar|n}}th distribution derivative of the Dirac delta function. This rule follows from 106 and 302.
|-
| 311
|<math> \frac{1}{x}</math>
|<math> -i\pi\sgn(\xi)</math>
|<math> -i\sqrt{\frac{\pi}{2}}\sgn(\omega)</math>
|<math> -i\pi\sgn(\omega)</math>
|Here {{math|sgn(''ξ'')}} is the [[sign function]]. Note that {{math|{{sfrac|1|''x''}}}} is not a distribution. It is necessary to use the [[Cauchy principal value]] when testing against [[Schwartz functions]]. This rule is useful in studying the [[Hilbert transform]].
|-
| 312
|<math>\begin{align}
&\frac{1}{x^n} \\
&:= \frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log |x|
\end{align}</math>
|<math> -i\pi \frac{(-i 2\pi \xi)^{n-1}}{(n-1)!} \sgn(\xi)</math>
|<math> -i\sqrt{\frac{\pi}{2}}\, \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)</math>
|<math> -i\pi \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)</math>
|{{math|{{sfrac|1|''x''<sup>''n''</sup>}}}} is the [[homogeneous distribution]] defined by the distributional derivative{{br}}<math>\frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log|x|</math>
|-
| 313
|<math> |x|^\alpha</math>
|<math> -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|2\pi\xi|^{\alpha+1}}</math>
|<math> \frac{-2}{\sqrt{2\pi}}\, \frac{\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} </math>
|<math> -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} </math>
|This formula is valid for {{math|0 > ''α'' > −1}}. For {{math|''α'' > 0}} some singular terms arise at the origin that can be found by differentiating 320. If {{math|Re ''α'' > −1}}, then {{math|{{abs|''x''}}<sup>''α''</sup>}} is a locally integrable function, and so a tempered distribution. The function {{math|''α'' ↦ {{abs|''x''}}<sup>''α''</sup>}} is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted {{math|{{abs|''x''}}<sup>''α''</sup>}} for {{math|''α'' ≠ −1, −3, ...}} (See [[homogeneous distribution]].)
|-
| <!-- Should we call it 313a ? Doesn't necessarily need a number, because it is a special case. -->
|<math> \frac{1}{\sqrt{|x|}} </math>
|<math> \frac{1}{\sqrt{|\xi|}} </math>
|<math> \frac{1}{\sqrt{|\omega|}}</math>
|<math> \frac{\sqrt{2\pi}}{\sqrt{|\omega|}} </math>
| Special case of 313.
|-
| 314
|<math> \sgn(x)</math>
|<math> \frac{1}{i\pi \xi}</math>
|<math> \sqrt{\frac{2}{\pi}} \frac{1}{i\omega } </math>
|<math> \frac{2}{i\omega }</math>
|The dual of rule 311. This time the Fourier transforms need to be considered as a [[Cauchy principal value]].
|-
| 315
|<math> u(x)</math>
|<math> \frac{1}{2}\left(\frac{1}{i \pi \xi} + \delta(\xi)\right)</math>
|<math> \sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right)</math>
|<math> \pi\left( \frac{1}{i \pi \omega} + \delta(\omega)\right)</math>
|The function {{math|''u''(''x'')}} is the Heaviside [[Heaviside step function|unit step function]]; this follows from rules 101, 301, and 314.
|-
| 316
|<math> \sum_{n=-\infty}^{\infty} \delta (x - n T)</math>
|<math> \frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( \xi -\frac{k }{T}\right)</math>
|<math> \frac{\sqrt{2\pi }}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)</math>
|<math> \frac{2\pi}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)</math>
|This function is known as the [[Dirac comb]] function. This result can be derived from 302 and 102, together with the fact that{{br}}<math>\begin{align}
& \sum_{n=-\infty}^{\infty} e^{inx} \\
= {}& 2\pi\sum_{k=-\infty}^{\infty} \delta(x+2\pi k)
\end{align}</math>{{br}}as distributions.
|-
| 317
|<math> J_0 (x)</math>
|<math> \frac{2\, \operatorname{rect}(\pi\xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} </math>
|<math> \sqrt{\frac{2}{\pi}} \, \frac{\operatorname{rect}\left(  \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
|<math> \frac{2\,\operatorname{rect}\left(\frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}}</math>
| The function {{math|''J''<sub>0</sub>(''x'')}} is the zeroth order [[Bessel function]] of first kind.
|-
| 318
|<math> J_n (x)</math>
|<math> \frac{2 (-i)^n T_n (2 \pi \xi) \operatorname{rect}(\pi \xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} </math>
|<math> \sqrt{\frac{2}{\pi}} \frac{ (-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
|<math> \frac{2(-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
| This is a generalization of 317. The function {{math|''J<sub>n</sub>''(''x'')}} is the {{mvar|n}}th order [[Bessel function]] of first kind. The function {{math|''T<sub>n</sub>''(''x'')}} is the [[Chebyshev polynomials|Chebyshev polynomial of the first kind]].
|-
| 319
|<math> \log \left| x \right|</math>
|<math> -\frac{1}{2} \frac{1}{\left| \xi \right|} - \gamma \delta \left( \xi \right) </math>
|<math> -\frac{\sqrt\frac{\pi}{2}}{\left| \omega \right|} - \sqrt{2 \pi} \gamma \delta \left( \omega \right) </math>
|<math> -\frac{\pi}{\left| \omega \right|} - 2 \pi \gamma \delta \left( \omega \right) </math>
|{{mvar|γ}} is the [[Euler–Mascheroni constant]]. It is necessary to use a finite part integral when testing {{math|{{sfrac|1|{{abs|''ξ''}}}}}} or {{math|{{sfrac|1|{{abs|''ω''}}}}}}against [[Schwartz functions]]. The details of this might change the coefficient of the delta function.
|- 
| 320
|<math> \left( \mp ix \right)^{-\alpha}</math>
|<math> \frac{\left(2\pi\right)^\alpha}{\Gamma\left(\alpha\right)}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha-1} </math>
|<math> \frac{\sqrt{2\pi}}{\Gamma\left(\alpha\right)}u\left(\pm\omega\right)\left(\pm\omega\right)^{\alpha-1} </math>
|<math> \frac{2\pi}{\Gamma\left(\alpha\right)}u\left(\pm\omega\right)\left(\pm\omega\right)^{\alpha-1} </math>
|This formula is valid for {{math|1 > ''α'' > 0}}. Use differentiation to derive formula for higher exponents. {{mvar|u}} is the Heaviside function.
|}