Separation of variables and Fourier transformation

Question

I know there's another question very similar to this argument. In the book Probabilità e Modelli Aleatori of Enzo Orsingher, at page 134, it shows that the transition function of an absorbing Brownian Motion is the solution of the following problem: $$ \left\{ \begin{array}{rl} \frac{du}{dt}=\frac{d^2u}{2dx^2} & x < a,t>0 \\ u(t_0,x,t_0,x_0)=\delta(x-x_0) \\ u(t,a,t_0,x_0)=0\\ \lim_{x \to -\infty} u(t,x,t_0,x_0)=0 & \lim_{x \to -\infty} \frac{du}{dx}(t,x,t_0,x_0)=0 \end{array} \right. $$ To solve this problem, the books uses the separation of variables and rewrites the solution of the heat equation in this way: $$u(t,x)=\int_{-\infty}^{\infty}e^{-\frac{1}{2}\beta^2t} \{A(\beta)e^{i\beta x}+B(\beta)e^{-i\beta x}\}$$ I really don't understand how to obtain this form. I looked a lot on the web for an answer but I didn't find it. Every time I look for the separation of variables in the heat equation, the result has a different form.