Oscillator representation of picture-changing operators

A rigorous derivation of the oscillator representation for the conformal picture-changing operators $e^{-\varphi(z)}$, $e^{\varphi(z)}$, and $Y(z)$ is given. The representation is expressed in several different forms. In the bracket form, the question of the extent to which the $\delta$-function representation for these operators is valid is elucidated. The regularization of products of picturechanging operators is briefly discussed.