Limit of distributions

Examples

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

${\displaystyle f_t(x)=\frac{t}{1+t^2{x}^2}}$

Since, by integration by parts,

${\displaystyle ⟨f_t,\varphi ⟩=-{\int }_{-\mathrm{\infty }}^0\arctan (tx){\varphi }^{\prime }(x)dx-{\int }_0^{\mathrm{\infty }}\arctan (tx){\varphi }^{\prime }(x)dx,}$

we have: ${\displaystyle {\displaystyle \underset{t\to \mathrm{\infty }}{lim}⟨f_t,\varphi ⟩=⟨\pi {\delta }_0,\varphi ⟩}}$. That is, the limit of ${\displaystyle f_t}$ as ${\displaystyle t\to \mathrm{\infty }}$ is ${\displaystyle \pi {\delta }_0}$.

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

One has

${\displaystyle (x-i0{)}^{-1}-(x+i0{)}^{-1}=2\pi i{\delta }_0.}$

Let ${\displaystyle {\mathrm{\Gamma }}_N=[-N-1/2,N+1/2{]}^2}$ be the rectangle with positive orientation, with an integer N. By the residue formula,

${\displaystyle I_N\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{{\mathrm{\Gamma }}_N}\hat{\varphi }(z)\pi \cot (\pi z)dz=2\pi i\sum _{-N}^N\hat{\varphi }(n).}$

On the other hand,

${\displaystyle \begin{array}{cc}{\int }_{-R}^R\hat{\varphi }(\xi )\pi \cot (\pi \xi )d& ={\int }_{-R}^R{\int }_0^{\mathrm{\infty }}\varphi (x)e^{-2\pi Ix\xi }dxd\xi +{\int }_{-R}^R{\int }_{-\mathrm{\infty }}^0\varphi (x)e^{-2\pi Ix\xi }dxd\xi \\ & =⟨\varphi ,\cot (\cdot -i0)-\cot (\cdot -i0)⟩\end{array}}$

Oscillatory integral