### Abstract

In polar coordinates, if , the expression , where is the Laplace–Beltrami operator, is the Laplacian of in the sense of the functions in . It is the Laplacian of in the sense of the functions in if and only if its extension by continuity in exists and is twice differentiable in . Otherwise, in particular, if is a singular function in , its Laplacian must be taken in the sense of the distributions. Now, this expression is generally considered as the Laplacian of in whatever 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 , the latter is generally considered as solutions in , which leads to contradictions with the theory of distributions. They are solved by introducing the algebra of the real-valued regular singular functions, where , is a non-negative real, and with the help of the operator Pf. These functions as also their Laplacian , which are defined in , define in the distributions and called pseudofunctions. We first show that the general solutions and to the Laplace and Helmoltz equations in are solutions in , not of these equations, but to the equations and , which can thus be regarded as extensions of the former in . Then we show that and , or more exactly the distributions and , are distribution solutions in to the Poisson and Helmoltz equations, and , where and are linear combinations of partial derivatives of the Dirac mass at the origin . The confusion between the Laplacians in and in , 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 and , and hence to take no account of the noncommutation of the operators Pf. and .

Received 15 October 2009
Accepted 20 January 2010
Published online 14 May 2010

Article outline:

I. INTRODUCTION
II. LAPLACIANS IN POLAR COORDINATES, LAPLACE, AND HELMOLTZ EQUATIONS
III. REGULAR SINGULAR FUNCTIONALGEBRA
A. Algebra of regular singular functions, “finite parts,” and pseudofunctions
1. Regular singular functionalgebra
2. Finite parts
3. Pseudofunctions
B. First problems of commutation
C. The operators and
D. Laplacian in of , where is a function in
E. Laplacian in of , where is a function of
IV. APPLICATIONS: PSEUDO-LAPLACE AND PSEUDO-HELMOLTZ EQUATIONS: DISTRIBUTION SOLUTIONS TO THE POISSON AND HELMOLTZ EQUATIONS

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