- #1

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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(k

_{1}, k

_{2}) and similarly for v(x,y) → γ(k

_{1}, k

_{2})

However, applying the cauchy riemann equations necessarily sets k

_{2}= i k

_{1}

and

which implies that the fourier transform of the full analytic function is simply

μ(k

_{1}, k

_{2}) =μ(k

_{1})δ(k

_{2}- ik

_{1}) = iγ(k

_{1}, k

_{2}) = iγ(k

_{1})δ(k

_{2}- ik

_{1})

(the δ(k

_{2}- ik

_{1}) 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)