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In [[mathematics]], the Fourier '''sine and cosine transforms''' are forms of the [[Fourier transform|Fourier integral transform]] that do not use [[complex number]]s. They are the forms originally used by [[Joseph Fourier]] and are still preferred in some applications, such as [[signal processing]] or [[statistics]].

==Definition==

The '''Fourier sine transform''' of <math> f (t) </math>, sometimes denoted by either <math> {\hat f}^s </math> or <math> {\mathcal F}_s (f) </math>, is

:<math> 2 \int\limits_{-\infty}^\infty f(t)\sin\,{2\pi \nu t} \,dt.</math>

If <math> t </math> means time, then <math> \nu</math> is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.

This transform is necessarily an [[odd function]] of frequency, i.e.,
:<math>  {\hat f}^s(\nu) = - {\hat f}^s(-\nu) </math> for all <math>\nu</math>.

The numerical factors in the [[Fourier transform]]s are defined uniquely only by their product.  Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has <math> L^2 </math> norm of <math> \frac 1 {\sqrt2} </math>.

The '''Fourier cosine transform''' of <math> f (t) </math>, sometimes denoted by either <math> {\hat f}^c </math> or <math> {\mathcal F}_c (f) </math>, is

:<math> 2 \int\limits_{-\infty}^\infty f(t)\cos\,{2\pi \nu t} \,dt.</math>

It is necessarily an [[even function]] of <math>\nu</math>, i.e.,
<math>  {\hat f}^c(\nu) =  {\hat f}^c(-\nu) </math> for all <math>\nu</math>.

Some authors<ref>[[Mary L. Boas]], ''[[Mathematical Methods in the Physical Sciences]]'', 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1</ref> only define the cosine transform for [[even function]]s of <math>t </math>, in which case its sine transform is zero.  Since cosine is also even,  a simpler formula can be used, <math> 4 \int\limits_0^\infty f(t)\cos\,{2\pi \nu t} \,dt.</math>  Similarly, if <math>f</math> is an [[odd function]], then the cosine transform is zero and the sine transform can be simplified to <math> 4 \int\limits_0^\infty f(t)\sin\,{2\pi \nu t} \,dt.</math>

==Fourier inversion==
The original function <math> f(t) </math> can be recovered from its transforms under the usual hypotheses, that <math> f </math> and both of its transforms should be absolutely integrable.  For more details on the different hypotheses, see [[Fourier inversion theorem]].

The inversion formula is<ref>{{cite book|last=[[Poincaré]]|first=Henri|title=Theorie analytique de la propagation de chaleur|year=1895|publisher=G. Carré|location=Paris|pages=pp. 108ff.|url=http://gallica.bnf.fr/ark:/12148/bpt6k5500702f/f115.image}}</ref>

:<math> f(t) = \int _0^\infty {\hat f}^c \cos (2\pi \nu t) d\nu + \int _0^\infty {\hat f}^s \sin (2\pi \nu t) d\nu,</math>

which has the advantage that all frequencies are positive and all quantities are real.   If the numerical factor 2 is left out of the definitions of the transforms, then the inversion formula is usually written as an integral over both negative and positive frequencies.

Using the addition formula for [[cosine]], this is sometimes rewritten as

:<math> \frac\pi2 (f(x+0)+f(x-0)) = \int _0^\infty \int_{-\infty}^\infty \cos \omega (t-x) f(t) dt d\omega, </math>

where <math> f(x+0) </math> denotes the one-sided [[limit]] of <math>f</math> as <math> x </math> approaches zero from above, and
<math> f(x-0) </math> denotes the one-sided limit of <math>f</math> as <math> x </math> approaches zero from below.

If the original function <math> f</math> is an [[even function]], then the sine transform is zero; if <math> f</math> is an [[odd function]], then the cosine transform is zero.  In either case, the inversion formula simplifies.

==Relation with complex exponentials==

The form of the [[Fourier transform]] used more often today is

:<math>
 \hat f(\nu)
 = \int\limits_{-\infty}^\infty f(t) e^{-2\pi i\nu t}\,dt.
</math>

Expanding the [[integrand]] by means of [[Euler's formula]] results in

:<math> = \int\limits_{-\infty}^\infty f(t)(\cos\,{2\pi\nu t} - i\,\sin{2\pi\nu t})\,dt,</math>

which may be written as the [[sum]] of two [[integral]]s

:<math> = \int\limits_{-\infty}^\infty f(t)\cos\,{2\pi \nu t} \,dt - i \int\limits_{-\infty}^\infty f(t)\sin\,{2\pi \nu t}\,dt,</math>

:<math> = \frac 12 {\hat f}^c (\nu) - \frac i2 {\hat f}^s (\nu). </math>

==See also==
*[[Discrete cosine transform]]
*[[Discrete sine transform]]

==References==

* Whittaker, Edmund, and James Watson, ''A Course in Modern Analysis'', Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211

<references />

[[Category:Integral transforms]]
[[Category:Fourier analysis]]