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

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

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

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

In n variables the homogeneity operator is given by

θ = ∑ k = 1 n x k ∂ ∂ x k . {\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)