User:Vanished user fijw983kjaslkekfhj45/Fourier Transform Tables

Definitions

Summary of popular forms of the Fourier transform

ordinary frequency ξ (hertz)	unitary	$\displaystyle {\hat {f}}_{1}(\xi )\ {\stackrel {\mathrm {def} }{=}}\ \int _{\mathbb {R} ^{n}}f(x)e^{-2\pi ix\cdot \xi }\,dx={\hat {f}}_{2}(2\pi \xi )=(2\pi )^{n/2}{\hat {f}}_{3}(2\pi \xi )$ $\displaystyle f(x)=\int _{\mathbb {R} ^{n}}{\hat {f}}_{1}(\xi )e^{2\pi ix\cdot \xi }\,d\xi \$
angular frequency ω (rad/s)	non-unitary	$\displaystyle {\hat {f}}_{2}(\nu )\ {\stackrel {\mathrm {def} }{=}}\int _{\mathbb {R} ^{n}}f(x)e^{-i\nu \cdot x}\,dx\ ={\hat {f}}_{1}\left({\frac {\nu }{2\pi }}\right)=(2\pi )^{n/2}\ {\hat {f}}_{3}(\nu )$ $\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\hat {f}}_{2}(\nu )e^{i\nu \cdot x}\,d\nu \$
	unitary	$\displaystyle {\hat {f}}_{3}(\nu )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{(2\pi )^{n/2}}}\int _{\mathbb {R} ^{n}}f(x)\ e^{-i\nu \cdot x}\,dx={\frac {1}{(2\pi )^{n/2}}}{\hat {f}}_{1}\left({\frac {\nu }{2\pi }}\right)={\frac {1}{(2\pi )^{n/2}}}{\hat {f}}_{2}(\nu )$ $\displaystyle f(x)={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbb {R} ^{n}}{\hat {f}}_{3}(\nu )e^{i\nu \cdot x}\,d\nu \$

Tables of important Fourier transforms

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency
	$\displaystyle f(x)\,$	$\displaystyle {\hat {f}}(\xi )=$ $\displaystyle \int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx$	$\displaystyle {\hat {f}}(\nu )=$${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx$	$\displaystyle {\hat {f}}(\omega )=$ $\displaystyle \int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx$
Functional relationships
101	$\displaystyle a\cdot f(x)+b\cdot g(x)\,$	$\displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,$	$\displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,$	$\displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,$
102	$\displaystyle f(x-a)\,$	$\displaystyle e^{-2\pi ia\xi }{\hat {f}}(\xi )\,$	$\displaystyle e^{-ia\nu }{\hat {f}}(\nu )\,$	$\displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,$
103	$\displaystyle e^{2\pi iax}f(x)\,$	$\displaystyle {\hat {f}}\left(\xi -a\right)\,$	$\displaystyle {\hat {f}}(\nu -2\pi a)\,$	$\displaystyle {\hat {f}}(\omega -2\pi a)\,$
104	$\displaystyle f(ax)\,$	$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,$	$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,$	$\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,$
105	$\displaystyle {\hat {f}}(x)\,$	$\displaystyle f(-\xi )\,$	$\displaystyle f(-\nu )\,$	$\displaystyle 2\pi f(-\omega )\,$
106	$\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,$	$\displaystyle (2\pi i\xi )^{n}{\hat {f}}(\xi )\,$	$\displaystyle (i\nu )^{n}{\hat {f}}(\nu )\,$
107	$\displaystyle x^{n}f(x)\,$	$\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\hat {f}}(\xi )}{d\xi ^{n}}}\,$	$\displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\nu )}{d\nu ^{n}}}$	$\displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}$
108	$\displaystyle (f*g)(x)\,$	$\displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,$	$\displaystyle {\sqrt {2\pi }}{\hat {f}}(\nu ){\hat {g}}(\nu )\,$	$\displaystyle {\hat {f}}(\omega ){\hat {g}}(\omega )\,$
109	$\displaystyle f(x)g(x)\,$	$\displaystyle ({\hat {f}}*{\hat {g}})(\xi )\,$	$\displaystyle ({\hat {f}}*{\hat {g}})(\nu ) \over {\sqrt {2\pi }}\,$	$\displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\omega )\,$
110	For $\displaystyle f(x)\,$ a purely real	$\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,$	$\displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,$	$\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,$
111	For $\displaystyle f(x)\,$ a purely real even function	$\displaystyle {\hat {f}}(\nu )$, $\displaystyle {\hat {f}}(\xi )$ and $\displaystyle {\hat {f}}(\omega )\,$ are purely real even functions.
112	For $\displaystyle f(x)\,$ a purely real odd function	$\displaystyle {\hat {f}}(\nu )$, $\displaystyle {\hat {f}}(\xi )$ and $\displaystyle {\hat {f}}(\omega )$ are purely imaginary odd functions.
Square-integrable functions
201	$\displaystyle \operatorname {rect} (ax)\,$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\xi }{a}}\right)$	$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} \left({\frac {\nu }{2\pi a}}\right)$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)$
202	$\displaystyle \operatorname {sinc} (ax)\,$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\xi }{a}}\right)\,$	$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {rect} \left({\frac {\nu }{2\pi a}}\right)$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)$
203	$\displaystyle \operatorname {sinc} ^{2}(ax)$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\xi }{a}}\right)$	$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {tri} \left({\frac {\nu }{2\pi a}}\right)$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)$
204	$\displaystyle \operatorname {tri} (ax)$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,$	$\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{2\pi a}}\right)$	$\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)$
205	$\displaystyle e^{-ax}u(x)\,$	$\displaystyle {\frac {1}{a+2\pi i\xi }}$	$\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\nu )}}$	$\displaystyle {\frac {1}{a+i\omega }}$
206	$\displaystyle e^{-\alpha x^{2}}\,$	$\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}$	$\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\nu ^{2}}{4\alpha }}}$	$\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}$
207	$\displaystyle \operatorname {e} ^{-a|x|}\,$	$\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}$	$\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\nu ^{2}}}$	$\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}$
208	$\displaystyle {\frac {J_{n}(x)}{x}}\,$	$\displaystyle {\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi \xi )\,$   $\displaystyle \cdot \ {\sqrt {1-4\pi ^{2}\xi ^{2}}}\operatorname {rect} (\pi \xi )$	$\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\nu )\,$   $\displaystyle \cdot \ {\sqrt {1-\nu ^{2}}}\operatorname {rect} \left({\frac {\nu }{2}}\right)$	$\displaystyle {\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,$   $\displaystyle \cdot \ {\sqrt {1-\omega ^{2}}}\operatorname {rect} \left({\frac {\omega }{2}}\right)$
209	$\displaystyle \operatorname {sech} (ax)\,$	$\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)$	$\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\nu \right)$	$\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)$
Distributions
301	$\displaystyle 1$	$\displaystyle \delta (\xi )$	$\displaystyle {\sqrt {2\pi }}\cdot \delta (\nu )$	$\displaystyle 2\pi \delta (\omega )$
302	$\displaystyle \delta (x)\,$	$\displaystyle 1$	$\displaystyle {\frac {1}{\sqrt {2\pi }}}\,$	$\displaystyle 1$
303	$\displaystyle e^{iax}$	$\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)$	$\displaystyle {\sqrt {2\pi }}\cdot \delta (\nu -a)$	$\displaystyle 2\pi \delta (\omega -a)$
304	$\displaystyle \cos(ax)$	$\displaystyle {\frac {\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}$	$\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\nu -a)+\delta (\nu +a)}{2}}\,$	$\displaystyle \pi \left(\delta (\omega -a)+\delta (\omega +a)\right)$
305	$\displaystyle \sin(ax)$	$\displaystyle {\frac {\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}$	$\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\nu -a)-\delta (\nu +a)}{2i}}$	$\displaystyle -i\pi \left(\delta (\omega -a)-\delta (\omega +a)\right)$
306	$\displaystyle \cos(ax^{2})$	$\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)$	$\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	$\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$
307	$\displaystyle \sin(ax^{2})\,$	$\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)$	$\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	$\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$
308	$\displaystyle x^{n}\,$	$\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )\,$	$\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\nu )\,$	$\displaystyle 2\pi i^{n}\delta ^{(n)}(\omega )\,$
309	$\displaystyle {\frac {1}{x}}$	$\displaystyle -i\pi \operatorname {sgn}(\xi )$	$\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\nu )$	$\displaystyle -i\pi \operatorname {sgn}(\omega )$
310	$\displaystyle {\frac {1}{x^{n}}}:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|$	$\displaystyle -i\pi {\frac {(-2\pi i\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )$	$\displaystyle -i{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )$	$\displaystyle -i\pi {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )$
311	$\displaystyle |x|^{\alpha }\,$	$\displaystyle -2{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}$	$\displaystyle {\frac {-2}{\sqrt {2\pi }}}{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|\nu |^{\alpha +1}}}$	$\displaystyle -2{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|\nu |^{\alpha +1}}}$
312	$\displaystyle \operatorname {sgn}(x)$	$\displaystyle {\frac {1}{i\pi \xi }}$	$\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\nu }}\,$	$\displaystyle {\frac {2}{i\omega }}$
313	$\displaystyle u(x)$	$\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)$	$\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \nu }}+\delta (\nu )\right)$	$\displaystyle \pi \left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)$
314	$\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)$	$\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)$	$\displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\nu -{\frac {2\pi k}{T}}\right)$	$\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)$
315	$\displaystyle J_{0}(x)$	$\displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}$	$\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\operatorname {rect} \left(\displaystyle {\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}$	$\displaystyle {\frac {2\,\operatorname {rect} \left(\displaystyle {\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$
316	$\displaystyle J_{n}(x)$	$\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}$	$\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\nu )\operatorname {rect} \left(\displaystyle {\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}$	$\displaystyle {\frac {2(-i)^{n}T_{n}(\omega )\operatorname {rect} \left(\displaystyle {\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$

Remarks

101. Linearity

102. Shift in time domain

103. Shift in frequency domain, dual of 102

104. Scaling in the time domain. If $\displaystyle |a|\,$ is large, then $\displaystyle f(ax)\,$ is concentrated around 0 and $\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,$ spreads out and flattens.

105. Duality. Here ${\hat {f}}$ needs to be calculated using the same method as Fourier transform column. Results from swapping "dummy" variables of $\displaystyle x\,$ and $\displaystyle \xi \,$ or $\displaystyle \nu \,$ or $\displaystyle \omega \,$.

107. This is the dual of 106

108. The notation $\displaystyle f*g\,$ denotes the convolution of $\displaystyle f\,$ and $\displaystyle g\,$ — this rule is the convolution theorem

109. This is the dual of 108

110. Hermitian symmetry. $\displaystyle {\overline {z}}\,$ indicates the complex conjugate.

201. The rectangular pulse and the normalized sinc function, here defined as sinc(x) = sin(πx)/(πx)

202. Dual of rule 201. The rectangular function is an ideal low-pass filter, and the sinc function is the non-causal impulse response of such a filter.

203. The function tri(x) is the triangular function

204. Dual of rule 203.

205. The function u(x) is the Heaviside unit step function and a>0.

206. This shows that, for the unitary Fourier transforms, the Gaussian function exp(−αx²) is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α)>0.

207. For a>0. That is, the Fourier transform of a decaying exponential function is a Lorentzian function.

208. The functions J_n (x) are the n-th order Bessel functions of the first kind. The functions U_n (x) are the Chebyshev polynomial of the second kind. See 315 and 316 below.

209. Hyperbolic secant is its own Fourier transform

301. The distribution δ(ξ) denotes the Dirac delta function.

302. Dual of rule 301.

303. This follows from 103 and 301.

304. This follows from rules 101 and 303 using Euler's formula: $\displaystyle \cos(ax)=(e^{iax}+e^{-iax})/2.$

305. This follows from 101 and 303 using $\displaystyle \sin(ax)=(e^{iax}-e^{-iax})/(2i).$

308. Here, n is a natural number and $\displaystyle \delta ^{(n)}(\xi )$ is the 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 polynomials.

309. 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.

310. 1/xⁿ is the homogeneous distribution defined by the distributional derivative $\textstyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|$

311. If Re α > −1, then $|x|^{\alpha }$ is a locally integrable function, and so a tempered distribution. The function $\alpha \mapsto |x|^{\alpha }$ 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 $|x|^{\alpha }$ for α ≠ −2, −4, ... (See homogeneous distribution.)

312. The dual of rule 309. This time the Fourier transforms need to be considered as Cauchy principal value.

313. The function u(x) is the Heaviside unit step function; this follows from rules 101, 301, and 312.

314. 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.

315. The function J₀(x) is the zeroth order Bessel function of first kind.

316. This is a generalization of 315. The function J_n(x) is the n-th order Bessel function of first kind. The function T_n(x) is the Chebyshev polynomial of the first kind.