Engineering Tables/Fourier Transform Table 2

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	$g(t)\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,$	$G(\omega )\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,$	$G(f)\!\equiv$ $\int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,$
10	$\mathrm {rect} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{\|a\|}}\cdot \mathrm {sinc} \left({\frac {f}{a}}\right)$	The rectangular pulse and the normalized sinc function
11	$\mathrm {sinc} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {rect} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{\|a\|}}\cdot \mathrm {rect} \left({\frac {f}{a}}\right)\,$	Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12	$\mathrm {sinc} ^{2}(at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {tri} \left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{\|a\|}}\cdot \mathrm {tri} \left({\frac {f}{a}}\right)$	tri is the triangular function
13	$\mathrm {tri} (at)\,$	${\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)$	${\frac {1}{\|a\|}}\cdot \mathrm {sinc} ^{2}\left({\frac {f}{a}}\right)\,$	Dual of rule 12.
14	$e^{-\alpha t^{2}}\,$	${\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}$	${\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi f)^{2}}{\alpha }}}$	Shows that the Gaussian function $\exp(-\alpha t^{2})$ is its own Fourier transform. For this to be integrable we must have $\mathrm {Re} (\alpha )>0$ .
	$e^{iat^{2}}=\left.e^{-\alpha t^{2}}\right\|_{\alpha =-ia}\,$	${\frac {1}{\sqrt {2a}}}\cdot e^{-i\left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$	${\sqrt {\frac {\pi }{a}}}\cdot e^{-i\left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$	common in optics
	$\cos(at^{2})\,$	${\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	${\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)$
	$\sin(at^{2})\,$	${\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)$	$-{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)$
	$e^{-a\|t\|}\,$	${\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}$	${\frac {2a}{a^{2}+4\pi ^{2}f^{2}}}$	a>0
	${\frac {1}{\sqrt {\|t\|}}}\,$	${\frac {1}{\sqrt {\|\omega \|}}}$	${\frac {1}{\sqrt {\|f\|}}}$	the transform is the function itself
	$J_{0}(t)\,$	${\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$	J₀(t) is the Bessel function of first kind of order 0, rect is the rectangular function
	$J_{n}(t)\,$	${\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}$	${\frac {2(-i)^{n}T_{n}(2\pi f)\mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}$	it's the generalization of the previous transform; T_n (t) is the Chebyshev polynomial of the first kind.
	${\frac {J_{n}(t)}{t}}\,$	${\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,$ $\cdot \ {\sqrt {1-\omega ^{2}}}\mathrm {rect} \left({\frac {\omega }{2}}\right)$	${\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,$ $\cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f)$	U_n (t) is the Chebyshev polynomial of the second kind

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