Consider the Dirac delta function, $\delta(x)$, defined by the sampling property

$$\int_{-\infty}^{\infty} f(x) \delta\left(x-x_{0}\right) d x=f\left(x_{0}\right)$$

for any suitable function $f(x)$ and real constant $x_{0}$.

(a) Show that $\delta(\alpha x)=|\alpha|^{-1} \delta(x)$ for any non-zero $\alpha \in \mathbb{R}$.

(b) Show that $x \delta^{\prime}(x)=-\delta(x)$, where ${ }^{\prime}$ denotes differentiation with respect to $x$.

(c) Calculate

$$\int_{-\infty}^{\infty} f(x) \delta^{(m)}(x) d x$$

where $\delta^{(m)}(x)$ is the $m^{\text {th }}$derivative of the delta function.

(d) For

$$\gamma_{n}(x)=\frac{1}{\pi} \frac{n}{(n x)^{2}+1}$$

show that $\lim _{n \rightarrow \infty} \gamma_{n}(x)=\delta(x)$.

(e) Find expressions in terms of the delta function and its derivatives for

(i)

$$\lim _{n \rightarrow \infty} n^{3} x e^{-x^{2} n^{2}}$$

(ii)

$$\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{0}^{n} \cos (k x) d k .$$

(f) Hence deduce that

$$\lim _{n \rightarrow \infty} \frac{1}{2 \pi} \int_{-n}^{n} e^{i k x} d k=\delta(x)$$

[You may assume that

$$\left.\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi} \quad \text { and } \quad \int_{-\infty}^{\infty} \frac{\sin y}{y} d y=\pi .\right]$$