Functional calculus

{{Short description|Theory allowing one to apply mathematical functions to mathematical operators}}

{{Short description|Theory allowing one to apply mathematical functions to mathematical operators}}

{{no footnotes|date=September 2025}}

{{no footnotes|date=September 2025}}

In [[mathematics]], a '''functional calculus''' is a theory allowing one to apply [[mathematical function]]s to [[mathematical operator]]s.https://users.metu.edu.tr/baver/chapter5.pdf from Preface on page 8. It is now a branch (more accurately, several related areas) of the field of [[functional analysis]], connected with [[spectral theory]]. Historically, the term was synonymous with the [[calculus of variations]]; the latter term remains in extensive use in physics and engineering texts, whereas functional calculus develops the subject further with more mathematically careful, formal, abstract and precise articulations. The older usage is still visible in the [[functional derivative]], which is often called the variational derivative. There are several unrelated uses of the term "functional calculus": it is sometimes applied to types of [[functional equations]], and sometimes to systems of logic in [[predicate calculus]].

In [[mathematics]], a '''functional calculus''' is a theory allowing one to apply [[mathematical function]]s to [[mathematical operator]]s.https://users.metu.edu.tr/baver/chapter5.pdf from Preface on page 8. It is now a branch (more accurately, several related areas) of the field of [[functional analysis]], connected with [[spectral theory]]. Historically, the term was synonymous with the [[calculus of variations]]; the latter term remains in extensive use in physics and engineering texts, whereas functional calculus develops the subject further with more mathematically careful, formal, abstract and precise articulations. The older usage is still visible in the [[functional derivative]], which is often called the variational derivative.

There are several unrelated uses of the term "functional calculus": it is sometimes applied to types of [[functional equations]], and sometimes to systems of logic in [[predicate calculus]].

Some of the areas of mathematics that fall under the term "functional calculus" include:

Some of the areas of mathematics that fall under the term "functional calculus" include:

* The [[operational calculus]], a technique for solving [[differential equation]]s by converting them into [[polynomial]] equations. The central idea is to view integration and differentiation as [[operator (mathematics)|operators]] in their own right; differential equations then appear to have the algebraic form of polynomials in these operators.

* The [[operational calculus]], a technique for solving [[differential equation]]s by converting them into [[polynomial]] equations. The central idea is to view integration and differentiation as [[operator (mathematics)|operators]] in their own right; differential equations then appear to have the algebraic form of polynomials in these operators.

* [[Holomorphic functional calculus]], which attempts to extend the techniques commonly used to study [[holomorphic function]]s $f(z)$ to expressions $f(T)$ where $T$ is a [[matrix (mathematics)|matrix]] or [[linear operator]]. This includes the special case where $f(z)$ is a polynomial, leading to the "polynomial functional calculus".

* [[Holomorphic functional calculus]], which attempts to extend the techniques commonly used to study [[holomorphic function]]s $f(z)$ to expressions $f(T)$ where $T$ is a [[matrix (mathematics)|matrix]] or [[linear operator]]. This includes the special case where $f(z)$ is a polynomial, leading to the "polynomial functional calculus".