# Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$ [duplicate]

I have the following problem:

Be the equation:

$$u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw$$

Show that $u\rightarrow 0$ as $x\rightarrow \infty$, even when $e^{-iwx}$ does not falter if $x\rightarrow \infty$.

The problem gives the hint to use integration by parts. I was hoping you explain this problem or help me solve it, for your attention and any help: thank you very much.

• After completing squares you get $$e^{-\frac{x^2}{4kt}} \int_{-\infty}^\infty c(w)e^{-kt \left(w + \frac{ix}{4 (kt)^2}\right )^2}dw$$ the inner thing looks like Gaussian distribution of some function. – Santosh Linkha May 11 '14 at 16:33
• Crossposted to physics.stackexchange.com/q/112271/2451 – Qmechanic May 11 '14 at 19:38