Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 9 / c / Solution

Factor the Gaussian function as $e^{-(x-a{)}^2}=e^{-a^2}{e}^{2ax}{e}^{-x^2}$. Multiplying the two exponential series givesFor $a\ne 0,\pm 1/\sqrt{2}$, its first three nonzero terms areVanishing coefficients require separate treatment in this Taylor expansion of a translated Gaussian. If $a=0$, the first three nonzero terms are $\boxed{1-x^2+x^4/2}$. If $a=\pm 1/\sqrt{2}$, they areThe latter expression retains the cubic term because the quadratic coefficient is zero.