The linearly polarized real-argument Hermite–Gauss beam is investigated by the Fourier transform method. The complex power is obtained and the reactive power of the paraxial beam is found to be zero. The complex space source required for the full-wave generalization of the real-argument Hermite–Gauss beam is deduced. The resulting basic full real-argument Hermite–Gauss wave is determined. The real and the reactive powers of the full wave are evaluated. The reactive power of the basic full real-argument Hermite–Gauss wave is infinite, and the reasons for this singularity are described. The real power depends on $k{w}_0$ , m, and n, where k is the wavenumber, $w_0$ is the e-folding distance of the Gaussian part of the input distribution, and m and n are the mode numbers. The variation in the real power with respect to changes in $k{w}_0$ for specified m and n as well as with respect to changes in m and n for a specified $k{w}_0$ is examined.