In mathematics, the theta operator is a differential operator defined by[1]
\theta=z{d\overdz}.
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:
\theta(zk)=kzk, k=0,1,2,...
In n variables the homogeneity operator is given by
\theta=
| n | |
| \sum | |
| k=1 |
xk
| \partial | |
| \partialxk |
.
As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem)