Abstract: | The regular beam equations are solved analytically for the case of emission from an arbitrary surface in conditions of total space charge ( -mode) and in a given external magnetic field H (§2) for temperature-limited emission (T-mode), in an external magnetic field H (§3); and for emission with nonzero initial velocity (§4). The emitter is taken as the coordinate surface x1=0 in an orthogonal system x1 (i = =1,2,3), while the current density J and field on it are given functions j(x2, x3), (x2, x3. The solution is written as series in (x1) with coefficients dependent on x2, x3, determined from recurrence relations. For emission in the -mode and H 0, =1/3; for temperature-limited emission, =1/2; with nonzero initial velocity, =1. The results are extended to the case of a beam in the presence of a moving background of uniform density (5). |