Define the Fourier transform $\tilde{f}(k)$ of a function $f(x)$ that tends to zero as $|x| \rightarrow \infty$, and state the inversion theorem. State and prove the convolution theorem.

Calculate the Fourier transforms of

Hence show that

$$\int_{-\infty}^{\infty} \frac{\sin (b k) e^{i k x}}{k\left(a^{2}+k^{2}\right)} d k=\frac{\pi \sinh (a b)}{a^{2}} e^{-a x} \quad \text { for } \quad x>b$$

and evaluate this integral for all other (real) values of $x$.