Theta operator

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

$\theta =z{d \over dz}.$

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

$\theta (z^{k})=kz^{k},\quad k=0,1,2,\dots$

In n variables the homogeneity operator is given by

$\theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.$

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.