State the convolution theorem for Laplace transforms.

The temperature $T(x, t)$ in a semi-infinite rod satisfies the heat equation

$$\frac{\partial^{2} T}{\partial x^{2}}=\frac{1}{k} \frac{\partial T}{\partial t}, \quad x \geq 0, t \geq 0$$

and the conditions $T(x, 0)=0$ for $x \geq 0, T(0, t)=f(t)$ for $t \geq 0$ and $T(x, t) \rightarrow 0$ as $x \rightarrow \infty$. Show that

$$T(x, t)=\int_{0}^{t} f(\tau) G(x, t-\tau) d \tau$$

where

$$G(x, t)=\sqrt{\frac{x^{2}}{4 \pi k t^{3}}} e^{-x^{2} / 4 k t}$$