# A quick question concerning error function

Why $$\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty}e^{tx}e^{-x^{2}/2}dx$$

equals to $$e^{t^{2}/2}$$ ?

I know it is error function. but I just do not have any basic knowledge about error function and do not know how to derive it.

• "I know it is error function": what ? – Yves Daoust Oct 17 '18 at 11:59

$$e^{tx-x^{2}/2}=e^{-(x-t)^{2}/2} e^{t^{2}}$$, Make the substitution $$y=x-t$$ to evaluate the integral.