Fourier transform of function of a complex variable

[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)]e^{ik(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)

 

[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.