Chapter 10. Fourier Transforms and the Dirac Delta Function

Vector Spaces in Physics 8/6/2015 10 - 1 Chapter 10. Fourier Transforms and the Dirac Delta Function A. The Fourier transform . The Fourier -series expansions which we have discussed are valid for functions either defined over a finite range (/ 2/ 2Tt T , for instance) or extended to all values of time as a periodic Function . This does not cover the important case of a single, isolated pulse. But we can approximate an isolated pulse by letting the boundaries of the region of the Fourier series recede farther and farther away towards , as shown in figure 10-1. We will now outline the corresponding mathematical limiting process. It will transform the Fourier series, a superposition of sinusoidal waves with discrete frequencies n, into a superposition of a continuous spectrum of frequencies.

Properties of the delta function By making a change of variable one can define the delta function in a more general way, so that the special point where it diverges is x = a (rather than x=0): x) g(x) Figure 10-4. The Gaussian function, becoming a delta function in the limit 0 . x a 1/a f(x) Figure 10-3. Rectangular function,

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  Chapter, Functions, Chapter 10, Delta, Transform, Fourier, Carid, Delta function, Fourier transforms and the dirac delta function

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