Theta operator

Short description: Mathematical operator

In mathematics, the theta operator is a differential operator defined by

[math]\displaystyle{ \theta = z {d \over dz}. }[/math]

This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:

[math]\displaystyle{ \theta (z^k) = k z^k,\quad k=0,1,2,\dots }[/math]

In n variables the homogeneity operator is given by

[math]\displaystyle{ \theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}. }[/math]

As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem)

See also

• Difference operator
• Delta operator
• Elliptic operator
• Fractional calculus
• Invariant differential operator
• Differential calculus over commutative algebras

Further reading

• Watson, G.N. (1995). A treatise on the theory of Bessel functions (Cambridge mathematical library ed., [Nachdr. der] 2. ed.). Cambridge: Univ. Press. ISBN 0521483913.