Regularization of a Distribution Holomorphic in a Parameter

We give sufficient conditions for regularizing a distribution of the form \( a(\sigma,\lambda)f(\lambda) \), where \( f(\lambda) \) is a distribution holomorphic in the parameter \( \lambda \), while \( a(\sigma,\lambda) \) is an infinitely differentiable function of \( \sigma \) outside some closed set \( N \) with power singularities of derivatives on \( N \) and holomorphic in \( \lambda \).