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The Exponential Fourier Transform

The Fourier transform is a reversible, linear transform with many uses. For or any function which may be real or complex, the Fourier transformis defined as

The associated inverse transform is

Not all functions have fourier transforms:does not for example. This function does not tend to zero astends to infinity.

The continuous exponential transform is related to the discrete transform via the identitybut has the advantage of transforming non periodic functions and being more suited to solving partial differential equations, and is more applicable in quantum mechanics where it allows the wavepackets representing particles, which are necessarily localized, to be decomposed into a range of frequencies with corresponding amplitudes that are non zero only in the region of the particle. Notice that bothandvary in the above equation betweenandIt operates in a way similar to the Laplace transform, transforming partial differential equations involving an unknown function  y which is to be found, where the original partial differential is in two variables, for example, into an algebraic equation, that can be solved for the fourier transform of the unknown function,The inverse transform can then be applied, usually using tables of standard fourier transforms, together with the convolution theory, which allows the inverse transform of products of fourier transforms.

The transform can be applied in space of any number of dimensions. In one dimension it is as above but in more than one dimensions it becomes

whereandaredimensional vectors.

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