An Elementary Algorithm for the Automatic Derivation and Proof of Tensor Product Identities via Computer Algebra

摘要

Tensor product, identities in two variables are quite; common in mathematics: Exponential, logarithmic, trigonometric. and hyperbolic functions all satisfy tensor product identities, and the binomial theorem is a familiar example of a tensor product identity for polynomial functions. This article presents a new elementary technique which can derive and prove all of these identities―automatically! This unified approach is based on the author's recent research on uniqueness theory tor dual asymptotic expansions and remainder theory for Taylor interpolation on two lines. These results provide a simple iterative algorithm which derives a tensor product from a closed-form expression using the author's asymptotic splitting operator, and a simple hyperbolic eigenfunction criterion which proves that the two forms are identically equal. The author has implemented these methods a.s a complete derivation and proof system in the Maple 8 computer algebra system. The Maple code, which is surprisingly brief, is included in its entirety. The article also includes numerous examples which illustrate a variety of novel techniques for deriving and proving tensor product identifies using this simple but effective system. (Maple is a registered trademark of Waterloo Maple Inc.)