User:AkanoToE/Tau

Table of some common angles in Tau radians

Several different units of angle measure are widely used, including degrees, radians, and gradians (gons):

1 full circle (turn) = 360 degrees =  τ radians  =  400 gons.

The following table shows the conversions and values for some common angles:

Turns	Degrees	Radians	Gradians	sine	cosine	tangent
0	0°	0	0g	0	1	0
⁠1/12⁠	30°	⁠ τ/12⁠	⁠33+1/3⁠g	⁠1/2⁠	⁠√3/2⁠	⁠√3/3⁠
⁠1/8⁠	45°	⁠ τ/8⁠	50g	⁠√2/2⁠	⁠√2/2⁠	1
⁠1/6⁠	60°	⁠ τ/6⁠	⁠66+2/3⁠g	⁠√3/2⁠	⁠1/2⁠	√3
⁠1/4⁠	90°	⁠ τ/4⁠	100g	1	0
⁠1/3⁠	120°	⁠ τ/3⁠	⁠133+1/3⁠g	⁠√3/2⁠	−⁠1/2⁠	−√3
⁠3/8⁠	135°	⁠3 τ/8⁠	150g	⁠√2/2⁠	−⁠√2/2⁠	−1
⁠5/12⁠	150°	⁠5 τ/12⁠	⁠166+2/3⁠g	⁠1/2⁠	−⁠√3/2⁠	−⁠√3/3⁠
⁠1/2⁠	180°	⁠ τ/2⁠	200g	0	−1	0
⁠7/12⁠	210°	⁠7 τ/12⁠	⁠233+1/3⁠g	−⁠1/2⁠	−⁠√3/2⁠	⁠√3/3⁠
⁠5/8⁠	225°	⁠5 τ/8⁠	250g	−⁠√2/2⁠	−⁠√2/2⁠	1
⁠2/3⁠	240°	⁠2 τ/3⁠	⁠266+2/3⁠g	−⁠√3/2⁠	−⁠1/2⁠	√3
⁠3/4⁠	270°	⁠3 τ/4⁠	300g	−1	0
⁠5/6⁠	300°	⁠5 τ/6⁠	⁠333+1/3⁠g	−⁠√3/2⁠	⁠1/2⁠	−√3
⁠7/8⁠	315°	⁠7 τ/8⁠	350g	−⁠√2/2⁠	⁠√2/2⁠	−1
⁠11/12⁠	330°	⁠11 τ/12⁠	⁠366+2/3⁠g	−⁠1/2⁠	⁠√3/2⁠	−⁠√3/3⁠
1	360°	τ	400g	0	1	0

Tables of important Fourier transforms

The following tables record some closed-form Fourier transforms. For functions f (x), g(x) and h(x) denote their Fourier transforms by f̂, ĝ, and ĥ respectively. Only the three most common conventions are included. It may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.

Functional relationships, one-dimensional

The Fourier transforms in this table may be found in Erdélyi (1954) or Kammler (2000, appendix).

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency	Remarks
	$f(x)\,$	${\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-\tau ix\xi }\,dx\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {\tau }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx\end{aligned}}$	Definition
101	$a\cdot f(x)+b\cdot g(x)\,$	$a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,$	$a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,$	$a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,$	Linearity
102	$f(x-a)\,$	$e^{-\tau ia\xi }{\hat {f}}(\xi )\,$	$e^{-ia\omega }{\hat {f}}(\omega )\,$	$e^{-ia\nu }{\hat {f}}(\nu )\,$	Shift in time domain
103	$f(x)e^{iax}\,$	${\hat {f}}\left(\xi -{\frac {a}{\tau }}\right)\,$	${\hat {f}}(\omega -a)\,$	${\hat {f}}(\nu -a)\,$	Shift in frequency domain, dual of 102
104	$f(ax)\,$	${\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,$	${\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,$	${\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,$	Scaling in the time domain. If |a| is large, then f (ax) is concentrated around 0 and ${\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,$ spreads out and flattens.
105	${\hat {f}}(x)\,$	$f(-\xi )\,$	$f(-\omega )\,$	$\tau f(-\nu )\,$	Duality. Here f̂ needs to be calculated using the same method as Fourier transform column. Results from swapping "dummy" variables of x and ξ or ω or ν.
106	${\frac {d^{n}f(x)}{dx^{n}}}\,$	$(\tau i\xi )^{n}{\hat {f}}(\xi )\,$	$(i\omega )^{n}{\hat {f}}(\omega )\,$	$(i\nu )^{n}{\hat {f}}(\nu )\,$
107	$x^{n}f(x)\,$	$\left({\frac {i}{\tau }}\right)^{n}{\frac {d^{n}{\hat {f}}(\xi )}{d\xi ^{n}}}\,$	$i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}$	$i^{n}{\frac {d^{n}{\hat {f}}(\nu )}{d\nu ^{n}}}$	This is the dual of 106
108	$(f*g)(x)\,$	${\hat {f}}(\xi ){\hat {g}}(\xi )\,$	${\sqrt {\tau }}{\hat {f}}(\omega ){\hat {g}}(\omega )\,$	${\hat {f}}(\nu ){\hat {g}}(\nu )\,$	The notation f ∗ g denotes the convolution of f and g — this rule is the convolution theorem
109	$f(x)g(x)\,$	$\left({\hat {f}}*{\hat {g}}\right)(\xi )\,$	${\frac {1}{\sqrt {\tau }}}\left({\hat {f}}*{\hat {g}}\right)(\omega )\,$	${\frac {1}{\tau }}\left({\hat {f}}*{\hat {g}}\right)(\nu )\,$	This is the dual of 108
110	For f (x) purely real	${\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,$	${\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,$	${\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,$	Hermitian symmetry. z indicates the complex conjugate.
111	For f (x) purely real and even	f̂ (ξ), f̂ (ω) and f̂ (ν) are purely real even functions.
112	For f (x) purely real and odd	f̂ (ξ), f̂ (ω) and f̂ (ν) are purely imaginary odd functions.
113	For f (x) purely imaginary	${\hat {f}}(-\xi )=-{\overline {{\hat {f}}(\xi )}}\,$	${\hat {f}}(-\omega )=-{\overline {{\hat {f}}(\omega )}}\,$	${\hat {f}}(-\nu )=-{\overline {{\hat {f}}(\nu )}}\,$	z indicates the complex conjugate.
114	${\overline {f(x)}}$	${\overline {{\hat {f}}(-\xi )}}$	${\overline {{\hat {f}}(-\omega )}}$	${\overline {{\hat {f}}(-\nu )}}$	Complex conjugation, generalization of 110 and 113
115	$f(x)\cos(ax)$	${\frac {{\hat {f}}\left(\xi -{\frac {a}{\tau }}\right)+{\hat {f}}\left(\xi +{\frac {a}{\tau }}\right)}{2}}$	${\frac {{\hat {f}}(\omega -a)+{\hat {f}}(\omega +a)}{2}}\,$	${\frac {{\hat {f}}(\nu -a)+{\hat {f}}(\nu +a)}{2}}$	This follows from rules 101 and 103 using Euler's formula: $\cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.$
116	$f(x)\sin(ax)$	${\frac {{\hat {f}}\left(\xi -{\frac {a}{\tau }}\right)-{\hat {f}}\left(\xi +{\frac {a}{\tau }}\right)}{2i}}$	${\frac {{\hat {f}}(\omega -a)-{\hat {f}}(\omega +a)}{2i}}$	${\frac {{\hat {f}}(\nu -a)-{\hat {f}}(\nu +a)}{2i}}$	This follows from 101 and 103 using Euler's formula: $\sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.$

Square-integrable functions, one-dimensional

The Fourier transforms in this table may be found in Campbell & Foster (1948), Erdélyi (1954), or Kammler (2000, appendix).

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency	Remarks
	$f(x)\,$	${\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-\tau ix\xi }\,dx\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {\tau }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx\end{aligned}}$
201	$\operatorname {rect} (ax)\,$	${\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\xi }{a}}\right)$	${\frac {1}{\sqrt {\tau a^{2}}}}\cdot \operatorname {sinc} \left({\frac {\omega }{\tau a}}\right)$	${\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\nu }{\tau a}}\right)$	The rectangular pulse and the normalized sinc function, here defined as sinc(x) = ⁠sin(τx/2)/(τx/2)⁠
202	$\operatorname {sinc} (ax)\,$	${\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\xi }{a}}\right)\,$	${\frac {1}{\sqrt {\tau a^{2}}}}\cdot \operatorname {rect} \left({\frac {\omega }{\tau a}}\right)$	${\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\nu }{\tau a}}\right)$	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. The function sinc(x) is the normalized sinc function.
203	$\operatorname {sinc} ^{2}(ax)$	${\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\xi }{a}}\right)$	${\frac {1}{\sqrt {\tau a^{2}}}}\cdot \operatorname {tri} \left({\frac {\omega }{\tau a}}\right)$	${\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\nu }{\tau a}}\right)$	The function tri(x) is the triangular function
204	$\operatorname {tri} (ax)$	${\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,$	${\frac {1}{\sqrt {\tau a^{2}}}}\cdot \operatorname {sinc} ^{2}\left({\frac {\omega }{\tau a}}\right)$	${\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{\tau a}}\right)$	Dual of rule 203.
205	$e^{-ax}u(x)\,$	${\frac {1}{a+\tau i\xi }}$	${\frac {1}{{\sqrt {\tau }}(a+i\omega )}}$	${\frac {1}{a+i\nu }}$	The function u(x) is the Heaviside unit step function and a > 0.
206	$e^{-\alpha x^{2}}\,$	${\sqrt {\frac {\tau }{2\alpha }}}\cdot e^{-{\frac {(\tau \xi /2)^{2}}{\alpha }}}$	${\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}$	${\sqrt {\frac {\tau }{2\alpha }}}\cdot e^{-{\frac {\nu ^{2}}{4\alpha }}}$	This shows that, for the unitary Fourier transforms, the Gaussian function e−αx2 is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α) > 0.
207	$e^{-i\alpha x^{2}}\,$	${\sqrt {\frac {\tau }{2\alpha }}}\cdot e^{i({\frac {(\tau \xi )^{2}}{4\alpha }}-{\frac {\tau }{8}})}$	${\frac {1}{\sqrt {2\alpha }}}\cdot e^{i({\frac {\omega ^{2}}{4\alpha }}-{\frac {\tau }{8}})}$	${\sqrt {\frac {\tau }{2\alpha }}}\cdot e^{i({\frac {\nu ^{2}}{4\alpha }}-{\frac {\tau }{8}})}$	This is known as the complex quadratic-phase sinusoid, or the "chirp" function.
208	$\operatorname {e} ^{-a|x|}\,$	${\frac {2a}{a^{2}+\tau ^{2}\xi ^{2}}}$	${\frac {2}{\sqrt {\tau }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}$	${\frac {2a}{a^{2}+\nu ^{2}}}$	For Re(a) > 0. That is, the Fourier transform of a two-sided decaying exponential function is a Lorentzian function.
209	$\operatorname {sech} (ax)\,$	${\frac {\tau }{2a}}\operatorname {sech} \left({\frac {\tau ^{2}}{4a}}\xi \right)$	${\frac {\sqrt {\tau }}{2a}}\operatorname {sech} \left({\frac {\tau }{4a}}\omega \right)$	${\frac {\tau }{2a}}\operatorname {sech} \left({\frac {\tau }{4a}}\nu \right)$	Hyperbolic secant is its own Fourier transform
210	$e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,$	${\frac {{\sqrt {\tau }}(-i)^{n}}{a}}e^{-{\frac {\tau ^{2}\xi ^{2}}{2a^{2}}}}H_{n}\left({\frac {\tau \xi }{a}}\right)$	${\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)$	${\frac {(-i)^{n}{\sqrt {\tau }}}{a}}e^{-{\frac {\nu ^{2}}{2a^{2}}}}H_{n}\left({\frac {\nu }{a}}\right)$	Hn is the nth-order Hermite polynomial. If a = 1 then the Gauss–Hermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see Hermite polynomial. The formula reduces to 206 for n = 0.

Distributions, one-dimensional

The Fourier transforms in this table may be found in Erdélyi (1954) or Kammler (2000, appendix).

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency	Remarks
	$f(x)\,$	${\begin{aligned}&{\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-\tau ix\xi }\,dx\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\omega )={\frac {1}{\sqrt {\tau }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\nu )=\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx\end{aligned}}$
301	$1$	$\delta (\xi )$	${\sqrt {\tau }}\cdot \delta (\omega )$	$\tau \delta (\nu )$	The distribution δ(ξ) denotes the Dirac delta function.
302	$\delta (x)\,$	$1$	${\frac {1}{\sqrt {\tau }}}\,$	$1$	Dual of rule 301.
303	$e^{iax}$	$\delta \left(\xi -{\frac {a}{\tau }}\right)$	${\sqrt {\tau }}\cdot \delta (\omega -a)$	$\tau \delta (\nu -a)$	This follows from 103 and 301.
304	$\cos(ax)$	${\frac {\delta \left(\xi -{\frac {a}{\tau }}\right)+\delta \left(\xi +{\frac {a}{\tau }}\right)}{2}}$	${\sqrt {\tau }}\cdot {\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}\,$	${\frac {\tau }{2}}\left(\delta (\nu -a)+\delta (\nu +a)\right)$	This follows from rules 101 and 303 using Euler's formula: $\cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.$
305	$\sin(ax)$	${\frac {\delta \left(\xi -{\frac {a}{\tau }}\right)-\delta \left(\xi +{\frac {a}{\tau }}\right)}{2i}}$	${\sqrt {\tau }}\cdot {\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}$	$-i{\frac {\tau }{2}}{\bigl (}\delta (\nu -a)-\delta (\nu +a){\bigr )}$	This follows from 101 and 303 using $\sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.$
306	$\cos \left(ax^{2}\right)$	${\sqrt {\frac {\tau }{2a}}}\cos \left({\frac {\tau ^{2}\xi ^{2}}{4a}}-{\frac {\tau }{8}}\right)$	${\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\tau }{8}}\right)$	${\sqrt {\frac {\tau }{2a}}}\cos \left({\frac {\nu ^{2}}{4a}}-{\frac {\tau }{8}}\right)$
307	$\sin \left(ax^{2}\right)\,$	$-{\sqrt {\frac {\tau }{2a}}}\sin \left({\frac {\tau ^{2}\xi ^{2}}{4a}}-{\frac {\tau }{8}}\right)$	${\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\tau }{8}}\right)$	$-{\sqrt {\frac {\tau }{2a}}}\sin \left({\frac {\nu ^{2}}{4a}}-{\frac {\tau }{8}}\right)$
308	$x^{n}\,$	$\left({\frac {i}{\tau }}\right)^{n}\delta ^{(n)}(\xi )\,$	$i^{n}{\sqrt {\tau }}\delta ^{(n)}(\omega )\,$	$\tau i^{n}\delta ^{(n)}(\nu )\,$	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.
	$\delta ^{(n)}(x)\,$	$(\tau i\xi )^{n}\,$	${\frac {(i\omega )^{n}}{\sqrt {\tau }}}\,$	$(i\nu )^{n}\,$	Dual of rule 308. δ(n)(ξ) is the nth distribution derivative of the Dirac delta function. This rule follows from 106 and 302.
309	${\frac {1}{x}}$	$-i{\frac {\tau }{2}}\operatorname {sgn}(\xi )$	$-i{\frac {\sqrt {\tau }}{2}}\operatorname {sgn}(\omega )$	$-i{\frac {\tau }{2}}\operatorname {sgn}(\nu )$	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.
310	${\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|\end{aligned}}$	$-i{\frac {\tau }{2}}{\frac {(-\tau i\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )$	$-i{\frac {\sqrt {\tau }}{2}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )$	$-i{\frac {\tau }{2}}{\frac {(-i\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )$	⁠1/xn⁠ is the homogeneous distribution defined by the distributional derivative ${\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|$
311	$|x|^{\alpha }\,$	$-{\frac {2\sin \left({\frac {\tau \alpha }{4}}\right)\Gamma (\alpha +1)}{|\tau \xi |^{\alpha +1}}}$	${\frac {-2}{\sqrt {\tau }}}\cdot {\frac {\sin \left({\frac {\tau \alpha }{4}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}$	$-{\frac {2\sin \left({\frac {\tau \alpha }{4}}\right)\Gamma (\alpha +1)}{|\nu |^{\alpha +1}}}$	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 ≠ −2, −4,... (See homogeneous distribution.)
	${\frac {1}{\sqrt {|x|}}}\,$	${\frac {1}{\sqrt {|\xi |}}}$	${\frac {1}{\sqrt {|\omega |}}}$	${\frac {\sqrt {\tau }}{\sqrt {|\nu |}}}$	Special case of 311.
312	$\operatorname {sgn}(x)$	${\frac {2}{i\tau \xi }}$	${\sqrt {\frac {1}{\tau }}}{\frac {2}{i\omega }}$	${\frac {2}{i\nu }}$	The dual of rule 309. This time the Fourier transforms need to be considered as a Cauchy principal value.
313	$u(x)$	${\frac {1}{2}}\left({\frac {2}{i\tau \xi }}+\delta (\xi )\right)$	${\frac {\sqrt {\tau }}{2}}\left({\frac {2}{i\tau \omega }}+\delta (\omega )\right)$	${\frac {\tau }{2}}\left({\frac {2}{i\tau \nu }}+\delta (\nu )\right)$	The function u(x) is the Heaviside unit step function; this follows from rules 101, 301, and 312.
314	$\sum _{n=-\infty }^{\infty }\delta (x-nT)$	${\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)$	${\frac {\sqrt {\tau }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {\tau k}{T}}\right)$	${\frac {\tau }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\nu -{\frac {\tau k}{T}}\right)$	This function is known as the Dirac comb function. This result can be derived from 302 and 102, together with the fact that $\sum _{n=-\infty }^{\infty }e^{inx}=\tau \sum _{k=-\infty }^{\infty }\delta (x+\tau k)$ as distributions.
315	$J_{0}(x)$	${\frac {2\,\operatorname {rect} \left({\frac {\tau \xi }{2}}\right)}{\sqrt {1-\tau ^{2}\xi ^{2}}}}$	${\frac {2}{\sqrt {\tau }}}\cdot {\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2\,\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}$	The function J0(x) is the zeroth order Bessel function of first kind.
316	$J_{n}(x)$	${\frac {2(-i)^{n}T_{n}(\tau \xi )\operatorname {rect} \left({\frac {\tau \xi }{2}}\right)}{\sqrt {1-\tau ^{2}\xi ^{2}}}}$	${\frac {2}{\sqrt {\tau }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2(-i)^{n}T_{n}(\nu )\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}$	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.
317	$\log \left|x\right|$	$-{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right)$	$-{\frac {\sqrt {\tau }}{2\left|\omega \right|}}-{\sqrt {\tau }}\gamma \delta \left(\omega \right)$	$-{\frac {\tau }{2\left|\nu \right|}}-\tau \gamma \delta \left(\nu \right)$	γ is the Euler–Mascheroni constant.
318	$\left(\mp ix\right)^{-\alpha }$	${\frac {\left(\tau \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1}$	${\frac {\sqrt {\tau }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}$	${\frac {\tau }{\Gamma \left(\alpha \right)}}u\left(\pm \nu \right)\left(\pm \nu \right)^{\alpha -1}$	This formula is valid for 1 > α > 0. Use differentiation to derive formula for higher exponents. u is the Heaviside function.

Two-dimensional functions

	Function	Fourier transform unitary, ordinary frequency	Fourier transform unitary, angular frequency	Fourier transform non-unitary, angular frequency	Remarks
400	$f(x,y)$	${\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\\&=\iint f(x,y)e^{-\tau i(\xi _{x}x+\xi _{y}y)}\,dx\,dy\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\\&={\frac {1}{\tau }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}$	${\begin{aligned}&{\hat {f}}(\nu _{x},\nu _{y})\\&=\iint f(x,y)e^{-i(\nu _{x}x+\nu _{y}y)}\,dx\,dy\end{aligned}}$	The variables ξx, ξy, ωx, ωy, νx, νy are real numbers. The integrals are taken over the entire plane.
401	$e^{-\tau \left(a^{2}x^{2}+b^{2}y^{2}\right)/2}$	${\frac {1}{|ab|}}e^{-{\frac {\tau }{2}}\left({\frac {\xi _{x}^{2}}{a^{2}}}+{\frac {\xi _{y}^{2}}{b^{2}}}\right)}$	${\frac {1}{\tau \cdot |ab|}}e^{-{\frac {1}{2\tau }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}$	${\frac {1}{|ab|}}e^{-{\frac {1}{2\tau }}\left({\frac {\nu _{x}^{2}}{a^{2}}}+{\frac {\nu _{y}^{2}}{b^{2}}}\right)}$	Both functions are Gaussians, which may not have unit volume.
402	$\operatorname {circ} \left({\sqrt {x^{2}+y^{2}}}\right)$	${\frac {J_{1}\left(\tau {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}$	${\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}$	${\frac {\tau J_{1}\left({\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}\right)}{\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}}$	The function is defined by circ(r) = 1 for 0 ≤ r ≤ 1, and is 0 otherwise. The result is the amplitude distribution of the Airy disk, and is expressed using J1 (the order-1 Bessel function of the first kind).
403	${\frac {1}{\sqrt {x^{2}+y^{2}}}}$	${\frac {1}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}$	${\frac {1}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}$	${\frac {\tau }{\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}}$	This is the Hankel transform of r-1, a 2-D Fourier "self-transform".
404	${\frac {i}{x+iy}}$	${\frac {1}{\xi _{x}+i\xi _{y}}}$	${\frac {1}{\omega _{x}+i\omega _{y}}}$	${\frac {\tau }{\nu _{x}+i\nu _{y}}}$