I am trying to do a fourier-transform of the function $$\psi(x,t=0) = \frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}e^{-\frac{x^2}{4\sigma^2}}e^{ik_0x}$$
My calculation is
$$\int_{-\infty}^\infty \psi(x,t=0)e^{-ikx}dx=\frac{1}{\sqrt{\sigma}(2\pi)^{1/4}} \int_{-\infty}^\infty e^{i(k_0-k)x-\frac{x^2}{4\sigma^2}}dx$$
Now I know that the following holds
$$\int_{-\infty}^\infty e^{-\frac{a}{2}x^2+bx}dx=\sqrt{\frac{2\pi}{a}}e^{\frac{b^2}{2a}}$$
which leads me to the fourier-transform of my function
$$\psi(x,t=0)=\frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}\sqrt{4\pi\sigma^2}e^{-(k_0-k)^2\sigma^2}=\pi^{1/4}\cdot 2^{3/4}\sqrt{\sigma}e^{-(k_0-k)^2\sigma^2}$$
but my book says the solution is $$\psi(x,t=0)=\pi^{1/4}\cdot 2^{3/4}e^{-(k_0-k)^2\sigma^2}$$ which leaves me confused. What am I doing wrong? Can anyone help me with this?