Can anyone point me to some material on applying the fourier transform to the case of an analytic function of one complex variable?(adsbygoogle = window.adsbygoogle || []).push({});

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)

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# Fourier transform of function of a complex variable

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