Integral representations of functions in the quantum disk. I

A $q$-analogue of the unit disk - the simplest homogeneous space of the quantum group $SU(1,1)$ is considered, $q$-analogues of the Cauchy–Green formula, the integral representation of eigen functions of the Laplace–Beltrami operator, the Green function for the Poisson equation and the formula of the Fourier transformation are given.