The diffusion equation,
$$\frac{\partial C(x,t)}{\partial t} = D\frac{\partial^2 C(x,t)}{\partial x^2} $$

has a product solution

$$ C = A(x) B(t).$$

By separation of variables, a time-dependant equation 

$$ \frac{dB}{dt} = - \lambda BD$$

and an $x$-dependent equation, 

$$ \frac{d^2A}{dx^2} = - \lambda A$$

are obtained, where $\lambda$ is the separation constant.

The x-dependent equation has solutions of the form

$$ \begin{array}{ccc} \cos \left( \sqrt{\lambda}x \right) & \textrm{and} & \sin \left( \sqrt{\lambda} x \right) \end{array} $$

from which a complex solution of the form $\textrm{e}^{-i\sqrt{\lambda}x}$ can be assumed.

The t-dependent equation has solutions of the form $\textrm{e}^{-D\lambda t}$. With the substitution, $\lambda = \omega^2$,

$$\begin{array}{ccc} n(\omega) \textrm{e}^{-D\omega^2 t} & \textrm{and} & m(\omega) \textrm{e} ^{-i \omega x}, \end{array} $$

are both solutions, where $n(\omega)$ and $m(\omega)$ are arbitrary functions of $\omega$ that can be combined to yield a new function, $k(\omega)$. By superposition, a solution may be obtained of the product form,

$$ k(\omega) \textrm{e}^{-D\omega^2 t} \textrm{e}^{-i\omega x}. $$

This solution may then be integrated over the entire range of $\omega$, $(-\infty \lt \omega \lt \infty)$ that yields a solution of the diffusion equation,

$$ C(x,t) = \int_{-\infty}^{\infty} k(\omega) \textrm{e}^{-i\omega x} \textrm{e}^{-D\omega^2 t} d\omega $$

that must then be made to satisfy the initial condition, (Eq. 2.4).