In mathematics, the theta operator is a
differential operator defined by
:
This is sometimes also called the homogeneity operator, because its
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s are the
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
s in ''z'':
:
In ''n'' variables the homogeneity operator is given by
:
As in one variable, the
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of ''θ'' are the spaces of
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
s. (
Euler's homogeneous function theorem
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
)
See also
*
Difference operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
*
Delta operator In mathematics, a delta operator is a shift-equivariant linear operator Q\colon\mathbb \longrightarrow \mathbb /math> on the vector space of polynomials in a variable x over a field \mathbb that reduces degrees by one.
To say that Q is shift-equi ...
*
Elliptic operator
*
Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
:D f(x) = \frac f(x)\,,
and of the integration ...
*
Invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fun ...
*
Differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this ...
References
Further reading
*{{cite book, last=Watson, first=G.N., title=A treatise on the theory of Bessel functions, year=1995, publisher=Univ. Press, location=Cambridge, isbn=0521483913, edition=Cambridge mathematical library ed.,
achdr. der2.
Differential operators