Theta operator

In mathematics, the theta operator is a differential operator defined by¹²

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

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

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

In n variables the homogeneity operator is given by

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

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