assignment Homework
Static Fields 2022 (5 years)
Remember that the delta function is defined so that
\[
\delta(x-a)=
\begin{cases}
0, &x\ne a\\
\infty, & x=a
\end{cases}
\]
Also:
\[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].
- Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
- Using the test function \(f(x)=3x^2\), find the value of
\[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\]
Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).