In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This notion generalizes a limit of a sequence of functions; a limit as a distribution may exist when a limit of functions does not.

The notion is a part of distributional calculus, a generalized form of calculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept of functions.

## Definition

Given a sequence of distributions $f_{i)$ , its limit $f$ is the distribution given by

$f[\varphi ]=\lim _{i\to \infty }f_{i}[\varphi ]$ for each test function $\varphi$ , provided that distribution exists. The existence of the limit $f$ means that (1) for each $\varphi$ , the limit of the sequence of numbers $f_{i}[\varphi ]$ exists and that (2) the linear functional $f$ defined by the above formula is continuous with respect to the topology on the space of test functions.

More generally, as with functions, one can also consider a limit of a family of distributions.

## Examples

A distributional limit may still exist when the classical limit does not. Consider, for example, the function:

$f_{t}(x)={t \over 1+t^{2}x^{2))$ Since, by integration by parts,

$\langle f_{t},\phi \rangle =-\int _{-\infty }^{0}\arctan(tx)\phi '(x)\,dx-\int _{0}^{\infty }\arctan(tx)\phi '(x)\,dx,$ we have: $\displaystyle \lim _{t\to \infty }\langle f_{t},\phi \rangle =\langle \pi \delta _{0},\phi \rangle$ . That is, the limit of $f_{t)$ as $t\to \infty$ is $\pi \delta _{0)$ .

Let $f(x+i0)$ denote the distributional limit of $f(x+iy)$ as $y\to 0^{+)$ , if it exists. The distribution $f(x-i0)$ is defined similarly.

One has

$(x-i0)^{-1}-(x+i0)^{-1}=2\pi i\delta _{0}.$ Let $\Gamma _{N}=[-N-1/2,N+1/2]^{2)$ be the rectangle with positive orientation, with an integer N. By the residue formula,

$I_{N}{\overset {\mathrm {def} }{=))\int _{\Gamma _{N)){\widehat {\phi ))(z)\pi \cot(\pi z)\,dz={2\pi i}\sum _{-N}^{N}{\widehat {\phi ))(n).$ On the other hand,

{\begin{aligned}\int _{-R}^{R}{\widehat {\phi ))(\xi )\pi \operatorname {cot} (\pi \xi )\,d&=\int _{-R}^{R}\int _{0}^{\infty }\phi (x)e^{-2\pi Ix\xi }\,dx\,d\xi +\int _{-R}^{R}\int _{-\infty }^{0}\phi (x)e^{-2\pi Ix\xi }\,dx\,d\xi \\&=\langle \phi ,\cot(\cdot -i0)-\cot(\cdot -i0)\rangle \end{aligned)) ## Oscillatory integral

 Main article: Oscillatory integral