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using notation more in line with the one used for the discrete Fourier transform |
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:''f''(''t'') = ∫ ''F''(''s'') e<sup>2π''ist''</sup> d''s''
for every real number ''t''.
As a rule of thumb: the more concentrated ''f''(''t'') is, the more spread out is ''F''(''s''). The only functions which coincide with their own Fourier transforms are the constant multiples of the function ''f''(''t'') = exp(-π''t''<sup>2</sup>). In a certain sense, this function therefore strikes the precise balance between being concentrated and being spread out.
Fourier transforms, and the closely related [[Laplace transform]]s are widely used in solving [[differential equations]]. The Fourier transform is compatible with [[derivative|differentiation]] in the following sense: if ''f''(''t'') is a differentiable function with Fourier transform ''F''(''s''), then the Fourier transform of its derivative is given by 2π''is'' ''F''(''s''). This can be used to transform differential equations into algebraic equations. Furthermore, the Fourier transform translates between [[convolution]] and multiplication of functions: if ''f''(''t'') and ''g''(''t'') are integrable functions with Fourier transforms ''F''(''s'') and ''G''(''s'') respectively, and if the convolution of ''f'' and ''g'' exists and is integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms ''F''(''s'') ''G''(''s''). If the the product ''f''(''t'') ''g''(''t'') is integrable, then the Fourier transform of this product is given by the convolution of ''F''(''s'') and ''G''(''s'').
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