Fourier transform of function of a complex variable

1. Nov 4, 2014

thoughtgaze

Can anyone point me to some material on applying the fourier transform to the case of an analytic function of one complex variable?

I've tried to generalize it myself, but I want to see if I'm overlooking some important things. I've started by writing the analytic function with

u + iv where u and v satisfy the cauchy riemann equations. I'm tempted to start by saying that to take the fourier transform of u + iv, simply take the fourier transform of u and v in the usual way for a function of two variables.

Usually you would have something like u(x,y) → μ(k1, k2) and similarly for v(x,y) → γ(k1, k2)

However, applying the cauchy riemann equations necessarily sets k2 = i k1
and
which implies that the fourier transform of the full analytic function is simply
μ(k1, k2) =μ(k1)δ(k2 - ik1) = iγ(k1, k2) = iγ(k1)δ(k2 - ik1)

(the δ(k2 - ik1) is just the dirac delta function, which I'm hoping is okay to use even with a complex argument.)

u + iv = ∫dk[2μ(k)]eik(x +iy) = ∫dkdx'dy'2e-i(k(x' - x +i(y'-y))) u(x,y)

μ(k) = ∫dxdy e-ik(x +iy) u(x,y)

2. Nov 5, 2014

thoughtgaze

Update: I am getting that a fourier transform is possible for the real part and imaginary part of an analytic function iff it is zero.