Irreducible Cartesian tensor expansions of scalar fields

Journal Article · Fri Aug 01 04:00:00 EDT 1975 · J. Math. Phys. (N.Y.), v. 16, no. 8, pp. 1550-1555

DOI:https://doi.org/10.1063/1.522722· OSTI ID:4146030

Moraal, H

It is shown how a scalar function V(parallel R + $Sigma$/sub i equals 1/ sup n/ a/sub i/parallel) of a sum of n + 1 vectors can be expanded as a multiple Cartesian tensor series in the vectors a/ sub i/. This expansion is a rearrangement of the multiple Taylor series expansion of such a function. In order to prove the fundamental theorem, generalized Cartesian Legendre polynomials are defined. The theorem is applied to the eigenfunctions of the Laplace operator and to inverse powers. The expansion of the latter type of function leads to forms involving generalized hypergeometric functions in several variables. As a special case, the Cartesian form of the multipole expansion of the electrostatic potential between two linear molecules is derived. A number of sum rules for hypergeometric functions and addition formulas for (standard and modified) spherical Bessel functions are proved by using a reduction property of the generalized Legendre polynomials. The case of the expansion of a tensorial function is also briefly discussed. (auth)

• Research Organization: Univ., Cologne
• Sponsoring Organization: USDOE
• NSA Number: NSA-33-008563
• OSTI ID: 4146030
• Journal Information: J. Math. Phys. (N.Y.), v. 16, no. 8, pp. 1550-1555, Journal Name: J. Math. Phys. (N.Y.), v. 16, no. 8, pp. 1550-1555; ISSN JMAPA
• Country of Publication: United States
• Language: English

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IRREDUCIBLE REPRESENTATIONS
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TENSORS