- #1

- 239

- 5

## Summary:

- In the definite integral of a delta function, how narrow can the interval be?

Given

\begin{equation}

\begin{split}

\int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y)

\end{split}

\end{equation}

where ##\epsilon > 0##

Is the following also true as ##\epsilon \rightarrow 0##

\begin{equation}

\begin{split}

\int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &\rightarrow \delta^{(2)}(x-y) f(x)

\\

&= f^{(2)}(y)

\end{split}

\end{equation}

If not, why?

\begin{equation}

\begin{split}

\int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y)

\end{split}

\end{equation}

where ##\epsilon > 0##

Is the following also true as ##\epsilon \rightarrow 0##

\begin{equation}

\begin{split}

\int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &\rightarrow \delta^{(2)}(x-y) f(x)

\\

&= f^{(2)}(y)

\end{split}

\end{equation}

If not, why?