Inversion formulae for the spherical mean in odd dimensions and the Euler-Poisson-Darboux equation

The paper contains a simple proof of the Finch-Patch-Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. Inversion of that transform is important for thermoacoustic tomography and represents a challenging mathematical problem. The argument relies on the idea of analytic continuation and known properties of the Erdélyi-Kober fractional integrals. The result is applied to the solution of the Cauchy problem for the Euler-Poisson-Darboux equation with initial data on the cylindrical surface.