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جدول تبدیل‌های فوریه مهم > توزیع‌ها، یک بعدی

|  | تابع | تبدیل فوریه فرکانس واحد، عادی | تبدیل فوریه فرکانس واحد، زاویه ای | تبدیل فوریه فرکانس زاویه ای غیر واحد | ملاحظات |
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|  |  f(x)\, |  {\begin{aligned}&{\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx\end{aligned}} |  {\begin{aligned}&{\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}} |  {\begin{aligned}&{\hat {f}}(\nu )=\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx\end{aligned}} |  |
| ۳۰۱ |  1 |  \delta (\xi ) |  {\sqrt {2\pi }}\cdot \delta (\omega ) |  2\pi \delta (\nu ) | The distribution δ(ξ) denotes the Dirac delta function. |
| ۳۰۲ |  \delta (x)\, |  1 |  {\frac {1}{\sqrt {2\pi }}}\, |  1 | Dual of rule 301. |
| ۳۰۳ |  e^{iax} |  \delta \left(\xi -{\frac {a}{2\pi }}\right) |  {\sqrt {2\pi }}\cdot \delta (\omega -a) |  2\pi \delta (\nu -a) | This follows from 103 and 301. |
| ۳۰۴ |  \cos(ax) |  {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}} |  {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}\, |  \pi \left(\delta (\nu -a)+\delta (\nu +a)\right) | This follows from rules 101 and 303 using Euler's formula:  \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}. |
| ۳۰۵ |  \sin(ax) |  {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}} |  {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}} |  -i\pi {\bigl (}\delta (\nu -a)-\delta (\nu +a){\bigr )} | This follows from 101 and 303 using  \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}. |
| ۳۰۶ |  \cos \left(ax^{2}\right) |  {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right) |  {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right) |  {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right) | This follows from 101 and 207 using  \cos(ax^{2})={\frac {e^{iax^{2}}+e^{-iax^{2}}}{2}}. |
| ۳۰۷ |  \sin \left(ax^{2}\right)\, |  -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right) |  {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right) |  -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right) | This follows from 101 and 207 using  \sin(ax^{2})={\frac {e^{iax^{2}}-e^{-iax^{2}}}{2i}}. |
| ۳۰۸ |  x^{n}\, |  \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )\, |  i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\, |  2\pi i^{n}\delta ^{(n)}(\nu )\, | Here, n is a natural number and δ(n)(ξ) is the nth distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all polynomials. |
|  |  \delta ^{(n)}(x)\, |  (2\pi i\xi )^{n}\, |  {\frac {(i\omega )^{n}}{\sqrt {2\pi }}}\, |  (i\nu )^{n}\, | Dual of rule 308. δ(n)(ξ) is the nth distribution derivative of the Dirac delta function. This rule follows from 106 and 302. |
| ۳۰۹ |  {\frac {1}{x}} |  -i\pi \operatorname {sgn}(\xi ) |  -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega ) |  -i\pi \operatorname {sgn}(\nu ) | Here sgn(ξ) is the sign function. Note that 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. |
| ۳۱۰ |  {\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|\end{aligned}} |  -i\pi {\frac {(-2\pi i\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi ) |  -i{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega ) |  -i\pi {\frac {(-i\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu ) | 1/xn is the homogeneous distribution defined by the distributional derivative  {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x| |
| ۳۱۱ |  |x|^{\alpha }\, |  -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}} |  {\frac {-2}{\sqrt {2\pi }}}\cdot {\frac {\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}} | This formula is valid for 0> α> −1. For α> 0 some singular terms arise at the origin that can be found by differentiating 318. If Re α> −1, then |x|α is a locally integrable function, and so a tempered distribution. The function α ↦ |x|α 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 |x|α for α ≠ −۲, −4,... (See homogeneous distribution.) |  |
|  |  {\frac {1}{\sqrt {|x|}}}\, |  {\frac {1}{\sqrt {|\xi |}}} |  {\frac {1}{\sqrt {|\omega |}}} |  {\frac {\sqrt {2\pi }}{\sqrt {|\nu |}}} | Special case of 311. |
| ۳۱۲ |  \operatorname {sgn}(x) |  {\frac {1}{i\pi \xi }} |  {\sqrt {\frac {2}{\pi }}}{\frac {1}{i\omega }} |  {\frac {2}{i\nu }} | The dual of rule 309. This time the Fourier transforms need to be considered as a Cauchy principal value. |
| ۳۱۳ |  u(x) |  {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right) |  {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right) |  \pi \left({\frac {1}{i\pi \nu }}+\delta (\nu )\right) | The function u(x) is the Heaviside unit step function; this follows from rules 101, 301, and 312. |
| ۳۱۴ |  \sum _{n=-\infty }^{\infty }\delta (x-nT) |  {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right) |  {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right) |  {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\nu -{\frac {2\pi k}{T}}\right) | This function is known as the Dirac comb function. This result can be derived from 302 and 102, together with the fact that  \sum _{n=-\infty }^{\infty }e^{inx}=2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k) as distributions. |
| ۳۱۵ |  J_{0}(x) |  {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}} |  {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}} |  {\frac {2\,\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}} | The function J0(x) is the zeroth order Bessel function of first kind. |
| ۳۱۶ |  J_{n}(x) |  {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}} |  {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}} |  {\frac {2(-i)^{n}T_{n}(\nu )\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}} | This is a generalization of 315. The function Jn(x) is the nth order Bessel function of first kind. The function Tn(x) is the Chebyshev polynomial of the first kind. |
| ۳۱۷ |  \log \left|x\right| |  -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right) |  -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta \left(\omega \right) |  -{\frac {\pi }{\left|\nu \right|}}-2\pi \gamma \delta \left(\nu \right) | γ is the Euler–Mascheroni constant. |
| ۳۱۸ |  \left(\mp ix\right)^{-\alpha } |  {\frac {\left(2\pi \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1} |  {\frac {\sqrt {2\pi }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1} |  {\frac {2\pi }{\Gamma \left(\alpha \right)}}u\left(\pm \nu \right)\left(\pm \nu \right)^{\alpha -1} | This formula is valid for 1> α> 0. Use differentiation to derive formula for higher exponents. u is the Heaviside function. |