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We investigate an application of slowly varying functions (in sense ofKaramata) in the theory of Markov branching process. We treat the critical caseso that the infinitesimal generating function of the process has the infinitesecond moment, but it regularly varies with the remainder. We improve the BasicLemma of the theory of critical Markov branching process and refine known limitresults.

On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance