The solution is clearly real.
What if we consider the complex integral:
; where I assume a,b are real and z = x + iy
when x = y.
tends to zero, when x is sufficiently large
So if we integrate along the path from the origin out to (R + iR) and then around the arc (centred on the origin) from R+iR to we find, after taking R to the limit of infinity, that the solution is zero.