What's the integral of e^(-w^2/1000^2 + 2.pi.w.i.x) dw ?

Question

This is the inverse Fourier transform of a low-pass filter, done in order to find the convolution kernel that would give the same result as this filter, if applied in the time domain.

Answer 1

Break the integral in 2 and do each part.
You get
1/200000*√π erf(w) + e^w/(2πix) + C,
where erf is the error function.
You can find out more about the error
function on www.integrals.wolfram.com .

Answer 2

... does the answer to this unlock the mysteries of time travel?