Editing Fourier transform (section)

=== Inversion ===
Still with <math>\xi = \sigma+ i\tau</math>, if <math>\widehat f</math> is complex analytic for {{math|''a'' ≤ ''τ'' ≤ ''b''}}, then

<math display="block"> \int _{-\infty}^\infty \hat f (\sigma + ia) e^{ i 2\pi \xi t} \, d\sigma = \int _{-\infty}^\infty \hat f (\sigma + ib) e^{ i 2\pi \xi t} \, d\sigma </math>
by [[Cauchy's integral theorem]]. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.<ref>{{harvnb|Wiener|1949}}</ref>

Theorem: If {{math|1=''f''(''t'') = 0}} for {{math|''t'' < 0}}, and {{math|{{abs|''f''(''t'')}} < ''Ce''<sup>''a''{{abs|''t''}}</sup>}} for some constants {{math|''C'', ''a'' > 0}}, then
<math display="block"> f(t) = \int_{-\infty}^\infty \hat f(\sigma + i\tau) e^{i 2 \pi \xi t} \, d\sigma,</math>
for any {{math|''τ'' < −{{sfrac|''a''|2π}}}}.

This theorem implies the [[inverse Laplace transform#Mellin's inverse formula|Mellin inversion formula]] for the Laplace transformation,<ref name="Kolmogorov-Fomin-1999" />
<math display="block"> f(t) = \frac 1 {i 2\pi} \int_{b-i\infty}^{b+i\infty} F(s) e^{st}\, ds</math>
for any {{math|''b'' > ''a''}}, where {{math|''F''(''s'')}} is the Laplace transform of {{math|''f''(''t'')}}.

The hypotheses can be weakened, as in the results of Carleson and Hunt, to {{math|''f''(''t'') ''e''<sup>−''at''</sup>}} being {{math|''L''<sup>1</sup>}}, provided that {{mvar|f}} be of bounded variation in a closed neighborhood of {{mvar|t}} (cf. [[Dini test]]), the value of {{mvar|f}} at {{mvar|t}} be taken to be the [[arithmetic mean]] of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.<ref>{{harvnb|Champeney|1987|p=63}}</ref>

{{math|''L''<sup>2</sup>}} versions of these inversion formulas are also available.<ref>{{harvnb|Widder|Wiener|1938|p=537}}</ref>