Mathematical Identity
From QED
In mathematics, an identity is an equation that remains true regardless of the values of any variables in it.
One general method for proving that two expressions are identical is to compute a canonical form for both sides, that is, to snap everything into a form which is one-to-one with the actual expression that it represents.
For example:
- Any simple trigonometric relation may be put into a form using eix.
- Polynomials may be expanded.
When extending an integer identity to the complex or real numbers, we may use the fact that a polynomial expression of degree d may have at most d distinct zeros, unless it is identially zero. Thus the difference of two polynomial expressions of degree at most d agree at more than d points, they are identical.
Indeed, it turns out that almost all hypergeometric identities may be proven algorithmically, in a simple, routine fashion. This covers an incredible amount of ground, including many special functions, such as the Bessel functions, the Gamma function, the Error Function, the Elliptic Integrals, and the Orthogonal Polynomials (including the Trigonometric Functions).
Once these methods are understood, an insight, is, indeed, no longer required for a proof. It is even possible to generate identities from an originating expression automatically. Intuition is still vital when attempting to remember, understand, or apply these identities, but now much of the magic has disappeared. It is now a science.
The book A = B is a fascinating piece of work that delves into the automation of the proof process for hypergeometric identities. It includes several methods which should be of interest to the physicist.
This page is derived from the Plasmagicians page Category:Identities as of 01:50, 26 October 2006.
