Theta operator

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In mathematics, the theta operator is a differential operator defined by[1][2]

\theta = z {d \over dz}

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

\theta (z^k) = k z^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 polynomials.

See also[edit]


  1. ^ "Theta Operator - from Wolfram MathWorld". Retrieved 2013-02-16. 
  2. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. (2nd ed.). Hoboken: CRC Press. pp. 2976–2983. ISBN 1420035223. 

Further reading[edit]

  • Watson, G.N. (1995). A treatise on the theory of Bessel functions (Cambridge mathematical library ed., [Nachdr. der] 2. ed. ed.). Cambridge: Univ. Press. ISBN 0521483913.