theta operator

{{Short description|Mathematical operator}}

In mathematics, the theta operator is a differential operator defined by{{MathWorld |id=ThetaOperator |title=Theta Operator |access-date=2013-02-16}}{{cite book|last=Weisstein|first=Eric W.|title=CRC Concise Encyclopedia of Mathematics|year=2002|publisher=CRC Press|location=Hoboken|isbn=1420035223|pages=2976–2983|edition=2nd}}

: $\backslash theta\; =\; z\; \{d\; \backslash over\; dz\}.$

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

:$\backslash theta\; (z^k)\; =\; k\; z^k,\backslash quad\; k=0,1,2,\backslash dots$

In n variables the homogeneity operator is given by

:$\backslash theta\; =\; \backslash sum\_\{k=1\}^n\; x\_k\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_k\}.$

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