2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107297The general relation between the standard expansion coefficients and the beta function for the QCD coupling is exactly derived in a mathematically strict way. It is accordingly found that an infinite number of logarithmic terms are lost in the standard expansion with a finite order, and these lost terms can be given in a closed form. Numerical calculations, by a new matching-invariant coupling with the corresponding beta function to four-loop level, show that the new expansion converges much faster.6 pages, 2 figures, RevTex4 styleHigh Energy Physics - Theorytext