Im just stuck beacause I dont know if this integral in bounded, I was trying to make a change of variable but I cant get to anything:
(edited what need is that f is bounded for a fixed x) $$f(x,t)=\int_{-\infty}^{+\infty} \exp \left(-\frac{(x-y)^{2}-2tby}{4t} \right)\;dy$$
where $b$ is constants. According to me it is bounded, but I can't proof it. How should I proceed?
I think I have the proof: Since $$u(x,t)=\frac{1}{\sqrt{4\pi Dt}}f(x,t)=\frac{1}{\sqrt{4\pi Dt}}\int_{-\infty}^{+\infty}exp\left(-\frac{(x-y)^{2}}{4Dt}\right)exp\left(\frac{b}{2D}y\right)dy$$ is the solution of the heat equation in $\mathbb{R}$ with inicial condition $u(x,0)=exp\left(\frac{b}{2D}x\right)$. Therefore $u(x,t)$ is bounded for a fixed $x$.