The Fourier transform $\tilde{h}(k)$ of the function $h(x)$ is defined by

$$\tilde{h}(k)=\int_{-\infty}^{\infty} h(x) e^{-i k x} d x$$

(i) State the inverse Fourier transform formula expressing $h(x)$ in terms of $\widetilde{h}(k)$.

(ii) State the convolution theorem for Fourier transforms.

(iii) Find the Fourier transform of the function $f(x)=e^{-|x|}$. Hence show that the convolution of the function $f(x)=e^{-|x|}$ with itself is given by the integral expression

$$\frac{2}{\pi} \int_{-\infty}^{\infty} \frac{e^{i k x}}{\left(1+k^{2}\right)^{2}} d k$$