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The "expansion by regions" is a method of asymptotic expansion developed byBeneke and Smirnov in 1997. It expands the integrand according to the scalingprescriptions of a set of regions and integrates all expanded terms over thewhole integration domain. This method has been applied successfully to manycomplicated loop integrals, but a general proof for its correctness has stillbeen missing. This paper shows how the expansion by regions manages toreproduce the exact result correctly in an expanded form and clarifies theconditions on the choice and completeness of the considered regions. Ageneralized expression for the full result is presented that involvesadditional overlap contributions. These extra pieces normally yield scalelessintegrals which are consistently set to zero, but they may be needed dependingon the choice of the regularization scheme. While the main proofs and formulaeare presented in a general and concise form, a large portion of the paper isfilled with simple, pedagogical one-loop examples which illustrate thepeculiarities of the expansion by regions, explain its application and show howto evaluate contributions within this method.

Foundation and generalization of the expansion by regions