  We discuss replica analytic continuation using several simple models in order
to prove mathematically the validity of replica analysis, which is used in a
wide range of fields related to large scale complex systems. While replica
analysis consists of two analytical techniques, the replica trick (or replica
analytic continuation) and the thermodynamical limit (and/or order parameter
expansion), we focus our study on replica analytic continuation, which is the
mathematical basis of the replica trick. We apply replica analysis to solve a
variety of analytical models, and examine the properties of replica analytic
continuation. Based on the positive results for these models we propose that
replica analytic continuation is a robust procedure in replica analysis.
