The basic equations are given for a wiggler-pumped free-electron laser with an axial-guide magnetic field. It is found that the evolution of the electron-field system is determined by a controlling parameter which contains the contributions of the electron beam, wiggler field, axial-guide magnetic field, and the initial state of the system. The collective instability is discussed and the threshold value of the controfling parameter is obtained. Farthermore, the radiation field is expressed in the cases of W＜WT and W＞WT.