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Mathematics
Calculus
Evaluation of an improper integral leading to a delta function
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[QUOTE="jasonRF, post: 6320803, member: 192203"] You haven't given us enough information to make sense out of (8). What are ##d##, ##b## and ##\alpha##? Also, your first equation has no ##x## in it at all, only ##\tilde{x}## and ##\tilde{x}^\prime##. I don't actually want you to answer these questions, by the way. To me this looks like the kind of integral you get when are using Fourier analysis to solve PDEs. What you need to understand is the basic idea, then your specific integral will be up to you. Consider $$ \begin{eqnarray*} \int_{-\infty}^\infty \int_{-\infty}^\infty \, F(\beta) e^{i (k - \beta)x} d\beta\, dx & = & \int_{-\infty}^\infty F(\beta) \left( \int_{-\infty}^\infty \, e^{i (k - \beta)x} dx\right) \, d\beta \\ & = & \int_{-\infty}^\infty F(\beta) \, 2\pi \, \delta(k-\beta) \\ & = & 2\pi\, F(k). \end{eqnarray*} $$ In the first line I just swapped the order of integration. In the second line I just used the fact that the Fourier transform of ##1## is ##2\pi\, \delta## (more on that below). In the third line I used the sifting property of the delta function which holds as long as ##F## is continuous at ##k##; this sifting property is often how delta functions are [I]defined[/I] by most non-rigorous applied treatments. What steps are you having trouble with? Note that we are using generalized functions (distributions) such as the delta function, and as such the integral symbols do not mean the same thing that they did when you learned calculus. When we write ##\int e^{ikx} dx = 2\pi \delta(k)## it we are simply saying that in the sense of generalized functions the Fourier transform of ##1## is ##\delta##, we are not evaluating the integral as written (indeed, the integral as written diverges). EDIT: note that the example I gave is just a statement that when you do the Fourier transform then the inverse Fourier transform of a function ##F## that you get to the original function. You can actually arrive at that conclusion using no delta functions at all. Jason [/QUOTE]
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Mathematics
Calculus
Evaluation of an improper integral leading to a delta function
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