The Laplace transform of the homogeneous functions in $\mathbb{R}^n$

In the one-dimensional case the Laplace transform of power functions relates to tabular integrals. The multidimensional analogue of power functions are homogeneous functions $\theta \left(\tau \right)\left|t\right|^{\alpha }$, where $\alpha$ is the degree of homogeneity and $\theta\left(\tau \right) $ is a function on the unit sphere $S^{n-1}$. For the convergence of the integral, it is necessary to consider the region $\gamma$ lying inside some hemisphere. In calculating the Laplace transform of homogeneous functions, it is necessary to derive an explicit representation. This is achieved by using Fourier analysis on the sphere, as well as summing the integrals applying the kernel of the Fourier transform, which allows us to construct a simple analytic continuation of the hypergeometric functions that appear in the calculations. The article obtains the formulas for the Laplace transform of homogeneous functions for which $\theta\left(\tau \right) $ belongs to different functional spaces on the unit sphere with support $\gamma$.