Testwiki:Reference desk/Archives/Mathematics/2014 February 8

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February 8

Limit

A derivation I am following quotes the result $\underset{a\to 0}{\lim }{\left(2a{\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{h}}^2\left(\frac{x}{a}\right)\right)}^{-1}=\delta (x)$ But I don't see how his can be, the LHS is not integrable. Does anyone know how this is obtained? — Preceding unsigned comment added by 81.156.72.65 (talk) 18:48, 8 February 2014 (UTC)

This does not look right. Is it not meant to be Template:Math in place of Template:Math? There are many functions that will work in this construction, but one requirement is that the function's definite integral from Template:Math to Template:Math must be finite (ignoring the usual abuse of notation used with the Dirac delta function, since it is not a function). —Quondum 20:26, 8 February 2014 (UTC)

Yes I am certain the context requires that it is $\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{h}$ (it is the fourier transform of $\omega \mathrm{C}\mathrm{o}\mathrm{t}\mathrm{h}\left(\frac{\omega }{2T}\right)$ ) which concerns me. I can provide a reference [1] it is equation 4.5 (pp30), though in this instance the limit is trated quite sloppily, it appears closer to the version I have given here as eq4.1 on pp44 of the published notes - the book [2]. — Preceding unsigned comment added by 81.156.72.65 (talk) 23:31, 8 February 2014 (UTC)

I do not say this with the confidence of a mathematician experienced in limits (I'm neither), but this looks to me like a case where the equivalence cannot be stated without justifying how the limiting process is done. Sometimes there are ways of dealing with divergent integrals, but these are a bit of a black art, and can yield different results depending on precisely how it is done. In a general context, this case would be considered undefined. —Quondum 01:45, 9 February 2014 (UTC)