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

= 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 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 .  — 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)