Definitions for functional calculus
func·tion·al cal·cu·lus

Princeton's WordNetRate this definition:0.0 / 0 votes

1. predicate calculus, functional calculusnoun

   a system of symbolic logic that represents individuals and predicates and quantification over individuals (as well as the relations between propositions)

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1. functional calculus

   Functional calculus refers to a mathematical theory where certain types of functions are applied to operators. These functions may not be traditional algebraic functions. Instead, they might belong to a wide range of mathematical functions including exponential functions, power series, and others. Typically, the field of functional calculus studies ways to define a function of a linear operator or more generally a non-commutative function. This enables the replacement of complex individual operations with simpler operations on the function values. The basic idea is to extend the notion of applying a function to a number, such as 'the square of 2' or 'the sine of π/2', to applying a function to an operator, for example, 'the square of the differentiation operator' or 'the exponential of the multiplication-by-x operator'. The functional calculus has a wide range of applications in mathematics, particularly in linear algebra, spectral theory and quantum mechanics.

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1. Functional calculus

   In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x² and M an n×n matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation. The most immediate case is to apply polynomial functions to a square matrix, extending what has just been discussed. In the finite dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator T. This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let n be the finite dimension of the algebra of matrices, then {I, T, T²...Tn} is linearly dependent. So ∑ αi Ti = 0 for some scalars αi, not all equal to 0. This implies that the polynomial ∑ αi xi lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial m. Multiplying by a unit if necessary, we can choose m to be monic. When this is done, the polynomial m is precisely the minimal polynomial of T. This polynomial gives deep information about T. For instance, a scalar α is an eigenvalue of T if and only if α is a root of m. Also, sometimes m can be used to calculate the exponential of T efficiently.

Matched Categories

1. • Symbolic Logic

How to pronounce functional calculus?

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How to say functional calculus in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of functional calculus in Chaldean Numerology is: 8

2. Pythagorean Numerology

   The numerical value of functional calculus in Pythagorean Numerology is: 9

References

1. ^ Princeton's WordNet
   http://wordnetweb.princeton.edu/perl/webwn?s=functional calculus
2. ^ ChatGPT
   https://chat.openai.com
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=functional calculus