This paper studies the properties of the Fourier transform of the fuzzy function, and extends the classical Poisson integral formula on the half plane to the fuzzy case, obtaining the composition of the fuzzy set generated by a point in the complex field under the action of the fuzzy function. Further, we define and study the fuzzy Hilbert transform of fuzzy functions and their properties. We prove that when the fuzzy function degenerates to the classical case, the fuzzy Hilbert transform will degenerate to the classical Hilbert transform, which proves that the fuzzy Hilbert transform is an extension of classical transformations in the fuzzy function space. In addition, we point out and prove some properties of the fuzzy Hilbert transform. For some fuzzy functions that meet certain requirements, their fuzzy Hilbert transform is a fuzzy point on 0.