A remark on the laplace transform

The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function $\cos \sqrt{z}$. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of $\cos \sqrt{z}$ using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces.