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F(k) = 1/sqrt(2a*[pi]) * [inte] exp( -(ax^2+ikx) dx over infinite limits. i is sqrt(-1)

to do this, I complete the square getting

exp( -(sqrt(a)*(x +ik/(2a))^2 * exp(k^2 / (4a))

sticking this in the integral and integrating over x I get

F(k) = 1/sqrt(2a*[pi]) * exp(k^2 / 4a )

I like the k^2 / a part but the factor of 4 seems wrong as well as the sign of the exponent possibly.

exp(-a x^2) should transform to exp(k^2/a)?

Help with using completing the square to do this integral would be greatly appreciated.

I used [inte] exp(-y^2) = sqrt([pi]) from a table.