Transcription of Chapter 10. Fourier Transforms and the Dirac Delta Function
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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 . As a starting point we rewrite the Fourier series, equation 9-39, as follows: /2/2()1nnitnnTitnTf tC enCf t edtT (10-1) The only change we have made is to add, in the upper expression, a factor of n for later use; 11nnn is the range of the variable n for each step in the summation.
2 f t g e dit Inverse Fourier Transform (10-8) 1 2 g f t e dt it Fourier Transform (10-9) There are a lot of notable things about these relations. First, there is a great symmetry in the roles of time and frequency; a function is completely specified either by f(t) or by g( ). Describing a
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