I am out of my depth here, but have an idea.

The solution is clearly real.

What if we consider the complex integral:

; where I assume a,b are real and z = x + iy

Now

when x = y.

also

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.