CCAST, World Lab, Beijing 100080, Peoples R China
; Chinese Acad Sci, Inst High Energy Phys, Beijing 100039, Peoples R China
; MIT, Ctr Theoret Phys, Cambridge, MA 02139 USA
The 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 logarithrnic 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. (c) 2006 Elsevier B.V. All rights reserved.