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