 By J. F. James

ISBN-10: 0521004284

ISBN-13: 9780521004282

This new version of a winning textbook for undergraduate scholars in physics, laptop technological know-how and electric engineering describes vital modern rules in functional technology and knowledge know-how at an comprehensible point, illustrated with labored examples and copious diagrams. the sector is roofed commonly instead of intensive, and comprises references to extra prolonged works on numerous subject matters. This re-creation is a bit multiplied, and contains extra new fabric within the functions sections.

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Additional info for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering

Example text

For the most part the proofs are elementary. The art of practical Fourier-transforming is in the manipulation of functions 22 Useful properties and theorems using these theorems, rather than in doing extensive and tiresome elementary integrations. It is this, as much as anything, which makes Fourier theory such a powerful tool for the practical working scientist. In what follows, we assume: where '~' implies that F] and <1>] are a Fourier pair. 1) The shift theorem already mentioned in Chapter 1 has the following lemmas: F](x + a) ~ <1>](p)e2rripa F](x - a) ~ <1>](p)e-2rripa F] (x - a) + F] (x + a) ~ 2<1>] (p) cos 2npa x= -a x= a Fig.

In general two separate sources, two laser beams for example, although nominally of the same wavelength, will not be coherent and no interference pattern will be seen when they both shine on to a screens. This is why, to generate 7 8 Equivalent to apodising in optics, but with more flexibility. This is not strictly true: a very fast detector can 'see' the fringes, which are shifting very rapidly on the surface where they are formed. Exposures in nanoseconds or less are required, and the technology involved is fairly expensive.

The result is one point on the graph of C(x'). 1 The convolution theorem With the exception of Fourier's Inversion Theorem, the convolution theorem is the most astonishing result in Fourier theory. It is as follows: If C(x) is the convolution of Fj (x) with F2(X) then its Fourier pair, r(p) is the productofcpj (p) and cp2(P), the Fourier pairs of F j (x) and F2(x). 5) The applications of this theorem are manifold and profound. Its proof is elementary: by definition. Fourier transform both sides (and note that, because the limits are ±oo, x' is a dummy variable and can be replaced by any other symbol not already in use): Introduce a new variable y = x - x'.