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

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February 8

Limit

A derivation I am following quotes the result $\lim _{a\rightarrow 0}\left(2\,a\,\mathrm {Sinh} ^{2}\left({\frac {x}{a}}\right)\right)^{-1}=\delta (x)$ 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 cosh in place of sinh? There are many functions that will work in this construction, but one requirement is that the function's definite integral from −∞ to +∞ 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 $\mathrm {Sinh}$ (it is the fourier transform of $\omega \mathrm {Coth} \left({\frac {\omega }{2T}}\right)$ ) 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)