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Homework for Periodic Systems

  1. (Delta Function Approximations)

    Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

    Also: $$\int_{-\infty}^{\infty} \delta(x-a)\, dx =1$$.

    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 $\epsilon$ approached zero.

    2. 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 $\epsilon \rightarrow 0$.

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