I have developed an extension of the rapid expansion method (REM) for 3D time-domain controlled-source electromagnetic modeling that includes perfectly matched layers (PMLs) as the absorbing boundary. The REM solves the time-domain electric field by a weighted summation of the Chebyshev polynomials. The results are free of temporal dispersion and accurate to the Nyquist frequency, yet the domain of Chebyshev polynomials lacks an accurate absorbing boundary. I find that by introducing a fictitious magnetic field in the Chebyshev domain, the recursion of the Chebyshev polynomials obeys a discrete coupled wave equation, which shares a similarity with the propagation of EM waves in a lossless medium. The time and frequency components in the Chebyshev domain are derived based on the eigenvalues of the propagation matrix, and the PML theory designed for EM waves can be extended to the Chebyshev domain in a straightforward way. Numerical tests against analytical solution and spectral methods show an excellent agreement after PML solves the boundary problem in the Chebyshev domain, which demonstrates the accuracy of the REM algorithm and the usefulness of the PML absorbing boundary.