Editing Fourier transform (section)

=== Sine and cosine transforms ===
{{Main|Sine and cosine transforms}}

Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function {{mvar|f}} for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically<ref>{{harvnb|Chatfield|2004|p=113}}</ref>) {{mvar|λ}} by
<math display="block">f(t) = \int_0^\infty \bigl( a(\lambda ) \cos( 2\pi \lambda t) + b(\lambda ) \sin( 2\pi \lambda t)\bigr) \, d\lambda.</math>

This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions {{mvar|a}} and {{mvar|b}} can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):
<math display="block"> a (\lambda) = 2\int_{-\infty}^\infty f(t) \cos(2\pi\lambda t) \, dt</math>
and
<math display="block"> b (\lambda) = 2\int_{-\infty}^\infty f(t) \sin(2\pi\lambda t) \, dt. </math>

Older literature refers to the two transform functions, the Fourier cosine transform, {{mvar|a}}, and the Fourier sine transform, {{mvar|b}}.

The function {{mvar|f}} can be recovered from the sine and cosine transform using
<math display="block"> f(t) = 2\int_0 ^{\infty} \int_{-\infty}^{\infty} f(\tau) \cos\bigl( 2\pi \lambda(\tau-t)\bigr) \, d\tau \, d\lambda.</math>
together with trigonometric identities. This is referred to as Fourier's integral formula.<ref name="Kolmogorov-Fomin-1999" /><ref>{{harvnb|Fourier|1822|p=441}}</ref><ref>{{harvnb|Poincaré|1895|p=102}}</ref><ref>{{harvnb|Whittaker|Watson|1927|p=188}}</ref>