Laplacian in polar coordinates, regular singular function algebra, and theory of distributions

In polar coordinates, if f∊C2 (Rn/0), the expression ∂2f/∂r2+ (n−1) r−1 (∂f/∂r) + (1/r2) Δσf, where Δσ is the Laplace–Beltrami operator, is the Laplacian of f in the sense of the functions in Rn/0. It is the Laplacian of f in the sense of the functions in Rn if and only if its extension by continuity in Rn exists and f is twice differentiable in Rn. Otherwise, in particular, if f is a singular function in Rn, its Laplacian must be taken in the sense of the distributions. Now, this expression is generally considered as the Laplacian of f in Rn whatever f is. This is why, although the Laplace and Helmoltz equations are solved by substituting this expression of the Laplacian, and although half of the solutions are singular functions in R3, the latter is generally considered as solutions in R3, which leads to contradictions with the theory of distributions. They are solved by introducing the algebra of the real-valued regular singular functions u/rs, where u∊C∞ (Rn), s is a non-negative real, and with the help of the operator Pf. These functions as also their Laplacian Δ (u/rs), which are defined in Rn/0, define in Rn the distributions Pf. (u/rs) and Pf. Δ (u/rs) called pseudofunctions. We first show that the general solutions fL and fH to the Laplace and Helmoltz equations in R3/0 are solutions in R3, not of these equations, but to the equations Pf. ΔfL=0 and Pf. (Δ+k2) fH=0, which can thus be regarded as extensions of the former in R3. Then we show that fL and fH, or more exactly the distributions Pf. fL and Pf. fH, are distribution solutions in R3 to the Poisson and Helmoltz equations, ΔPf. fL=TL and (Δ+k2) Pf. fH=TH, where TL and TH are linear combinations of partial derivatives of the Dirac mass at the origin δ0. The confusion between the Laplacians in Rn/0 and in Rn, or/and between the Laplacians in the sense of the functions and in the sense of the distributions, amounts in terms of the distributions to identify the operators ΔPf. and Pf. Δ, and hence to take no account of the noncommutation of the operators Pf. and Δ.